Chapter 4 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

##

Worked example

break-even point

coincident lines

y

complementary angles

Two angles are complementary if the sum of the measures of their angles is 90 degrees.

consistent and inconsistent systems

Consistent system of equations is a system of equations with at least one solution; inconsistent system of equations is a system of equations with no solution.

cost function

C

x

x

determinant

Each square matrix has a real number associated with it called its determinant.

matrix

A matrix is a rectangular array of numbers arranged in rows and columns.

revenue

x

R(x

x

row-echelon form

A matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

solutions of a system of equations

all

solutions of a system of linear equations with three variables

square matrix

A square matrix is a matrix with the same number of rows and columns.

supplementary angles

Two angles are supplementary if the sum of the measures of their angles is 180 degrees.

system of linear equations

When two or more linear equations are grouped together, they form a system of linear equations.

system of linear inequalities

Two or more linear inequalities grouped together form a system of linear inequalities.

##

Key Concepts

• How to solve a system of linear equations by graphing. Step 1. Graph the first equation. Step 2. Graph the second equation on the same rectangular coordinate system. Step 3. Determine whether the lines intersect, are parallel, or are the same line. Step 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions. Step 5. Check the solution in both equations.

• How to solve a system of equations by substitution. Step 1. Solve one of the equations for either variable. Step 2. Substitute the expression from Step 1 into the other equation. Step 3. Solve the resulting equation. Step 4. Substitute the solution in Step 3 into either of the original equations to find the other variable. Step 5. Write the solution as an ordered pair. Step 6. Check that the ordered pair is a solution to both original equations.

• How to solve a system of equations by elimination. Step 1. Write both equations in standard form. If any coefficients are fractions, clear them. Step 2. Make the coefficients of one variable opposites.Decide which variable you will eliminate.Multiply one or both equations so that the coefficients of that variable are opposites. Step 3. Add the equations resulting from Step 2 to eliminate one variable. Step 4. Solve for the remaining variable. Step 5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. Step 6. Write the solution as an ordered pair. Step 7. Check that the ordered pair is a solution to both original equations. $\begin{array}{llccccccccccc} \mathrm{Choose the Most Convenient Method to Solve a System of Linear Equations} \\ \begin{array}{lcclccl} \underset{\mathrm{---------------}}{\mathrm{Graphing}} & & & \underset{\mathrm{---------------------}}{\mathrm{Substitution}} & & & \underset{\mathrm{---------------------}}{\mathrm{Elimination}} \\ \begin{array}{l} \mathrm{Use when you need a} \\ \mathrm{picture of the situation.} \\ \end{array} & & & \begin{array}{l} \mathrm{Use when one equation is} \\ \mathrm{already solved or can be} \\ \mathrm{easily solved for one} \\ \mathrm{variable.} \\ \end{array} & & & \begin{array}{l} \mathrm{Use when the equations are} \\ \mathrm{in standard form.} \\ \end{array} \\ \end{array} \\ \end{array}$

• How To Solve Applications with Systems of Equations Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose variables to represent those quantities. Step 4. Translate into a system of equations. Step 5. Solve the system of equations using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Cost function: The cost function is the cost to manufacture each unit times x, the number of units manufactured, plus the fixed costs.

• Revenue: The revenue function is the selling price of each unit times x, the number of units sold.

• Break-even point: The break-even point is when the revenue equals the costs.

• Linear Equation in Three Variables: A linear equation with three variables, where a, b, c, and d are real numbers and a, b, and c are not all 0, is of the form

Every solution to the equation is an ordered triple, $(x,y,z)$ that makes the equation true.

• How to solve a system of linear equations with three variables. Step 1. Write the equations in standard form If any coefficients are fractions, clear them. Step 2. Eliminate the same variable from two equations. Decide which variable you will eliminate. Work with a pair of equations to eliminate the chosen variable. Multiply one or both equations so that the coefficients of that variable are opposites. Add the equations resulting from Step 2 to eliminate one variable Step 3. Repeat Step 2 using two other equations and eliminate the same variable as in Step 2. Step 4. The two new equations form a system of two equations with two variables. Solve this system. Step 5. Use the values of the two variables found in Step 4 to find the third variable. Step 6. Write the solution as an ordered triple. Step 7. Check that the ordered triple is a solution to all three original equations.

• Matrix: A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with m rows and n columns has order $m\,\times\,n.$ The matrix on the left below has 2 rows and 3 columns and so it has order $2\,\times\,3.$ We say it is a 2 by 3 matrix. Each number in the matrix is called an element or entry in the matrix.

• Row Operations: In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix. Interchange any two rows Multiply a row by any real number except 0 Add a nonzero multiple of one row to another row

• Row-Echelon Form: For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

• How to solve a system of equations using matrices. Step 1. Write the augmented matrix for the system of equations. Step 2. Using row operations get the entry in row 1, column 1 to be 1. Step 3. Using row operations, get zeros in column 1 below the 1. Step 4. Using row operations, get the entry in row 2, column 2 to be 1. Step 5. Continue the process until the matrix is in row-echelon form. Step 6. Write the corresponding system of equations. Step 7. Use substitution to find the remaining variables. Step 8. Write the solution as an ordered pair or triple. Step 9. Check that the solution makes the original equations true.

• Determinant: The determinant of any square matrix $[\begin{array}{l} a\,b \\ c\,d \\ \end{array}],$ where a, b, c, and d are real numbers, is

• Expanding by Minors along the First Row to Evaluate a 3 × 3 Determinant: To evaluate a $3\,\times\,3$ determinant by expanding by minors along the first row, the following pattern:

• Sign Pattern: When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.

• Cramer’s Rule: For the system of equations $\mathrm{\{}\begin{array}{l} {a}_{1}x+{b}_{1}y={k}_{1} \\ {a}_{2}x+{b}_{2}y={k}_{2} \\ \end{array},$ the solution $(x,y)$ can be determined by Notice that to form the determinant D, we use take the coefficients of the variables.

• How to solve a system of two equations using Cramer’s rule. Step 1. Evaluate the determinant D, using the coefficients of the variables. Step 2. Evaluate the determinant ${D}_{x}.$ Use the constants in place of the x coefficients. Step 3. Evaluate the determinant ${D}_{y}.$ Use the constants in place of the y coefficients. Step 4. Find x and y. $x=\frac{{D}_{x}}{D},$$y=\frac{{D}_{y}}{D}.$ Step 5. Write the solution as an ordered pair. Step 6. Check that the ordered pair is a solution to both original equations. Step 7. Dependent and Inconsistent Systems of Equations: For any system of equations, where the value of the determinant $D=0,$$\begin{array}{lcclccl} \mathrm{Value of determinants} & & & \mathrm{Type of system} & & & \mathrm{Solution} \\ D=0\,\mathrm{and}\,{D}_{x},{D}_{y}\,\mathrm{and}\,{D}_{z}\,\mathrm{are all zero} & & & \mathrm{consistent and dependent} & & & \mathrm{infinitely many solutions} \\ D=0\,\mathrm{and}\,{D}_{x},{D}_{y}\,\mathrm{and}\,{D}_{z}\,\mathrm{are not all zero} & & & \mathrm{inconsistent} & & & \mathrm{no solution} \\ \end{array}$ Step 8. Test for Collinear Points: Three points $({x}_{1},{y}_{1}),$$({x}_{2},{y}_{2}),$ and $({x}_{3},{y}_{3})$ are collinear if and only if

• Solutions of a System of Linear Inequalities: Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the x, y coordinate system that includes all the points whose ordered pairs make the inequalities true.

• How to solve a system of linear inequalities by graphing. Step 1. Graph the first inequality.Graph the boundary line.Shade in the side of the boundary line where the inequality is true. Step 2. On the same grid, graph the second inequality.Graph the boundary line.Shade in the side of that boundary line where the inequality is true. Step 3. The solution is the region where the shading overlaps. Step 4. Check by choosing a test point.

##

Review Exercises

Determine Whether an Ordered Pair is a Solution of a System of Equations.

In the following exercises, determine if the following points are solutions to the given system of equations.

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

In the following exercises, without graphing determine the number of solutions and then classify the system of equations.

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

Solve Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

Solve Mixture Applications with Systems of Equations

For the following exercises, translate to a system of equations and solve.

Solve Interest Applications

For the following exercises, translate to a system of equations and solve.

Solve Systems of Equations with Three Variables

In the following exercises, determine whether the ordered triple is a solution to the system.

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

Solve Applications using Systems of Linear Equations with Three Variables

Write the Augmented Matrix for a System of Equations.

Write each system of linear equations as an augmented matrix.

Write the system of equations that that corresponds to the augmented matrix.

In the following exercises, perform the indicated operations on the augmented matrices.

Solve Systems of Equations Using Matrices

In the following exercises, solve each system of equations using a matrix.

Evaluate the Determinant of a 2 × 2 Matrix

In the following exercise, evaluate the determinate of the square matrix.

Evaluate the Determinant of a 3 × 3 Matrix

In the following exercise, find and then evaluate the indicated minors.

In the following exercise, evaluate each determinant by expanding by minors along the first row.

In the following exercise, evaluate each determinant by expanding by minors.

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s rule

Solve Applications Using Determinants

In the following exercises, determine whether the given points are collinear.

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

In the following exercises, determine whether each ordered pair is a solution to the system.

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

##

In the following exercises, solve the following systems by graphing.

In the following exercises, solve each system of equations. Use either substitution or elimination.

Solve the system of equations using a matrix.

Solve using Cramer’s rule.

In the following exercises, translate to a system of equations and solve.

Key Terms

break-even point

coincident lines

y

complementary angles

Two angles are complementary if the sum of the measures of their angles is 90 degrees.

consistent and inconsistent systems

Consistent system of equations is a system of equations with at least one solution; inconsistent system of equations is a system of equations with no solution.

cost function

C

x

x

determinant

Each square matrix has a real number associated with it called its determinant.

matrix

A matrix is a rectangular array of numbers arranged in rows and columns.

revenue

x

R(x

x

row-echelon form

A matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

solutions of a system of equations

all

solutions of a system of linear equations with three variables

square matrix

A square matrix is a matrix with the same number of rows and columns.

supplementary angles

Two angles are supplementary if the sum of the measures of their angles is 180 degrees.

system of linear equations

When two or more linear equations are grouped together, they form a system of linear equations.

system of linear inequalities

Two or more linear inequalities grouped together form a system of linear inequalities.

##

Key Concepts

• How to solve a system of linear equations by graphing. Step 1. Graph the first equation. Step 2. Graph the second equation on the same rectangular coordinate system. Step 3. Determine whether the lines intersect, are parallel, or are the same line. Step 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions. Step 5. Check the solution in both equations.

• How to solve a system of equations by substitution. Step 1. Solve one of the equations for either variable. Step 2. Substitute the expression from Step 1 into the other equation. Step 3. Solve the resulting equation. Step 4. Substitute the solution in Step 3 into either of the original equations to find the other variable. Step 5. Write the solution as an ordered pair. Step 6. Check that the ordered pair is a solution to both original equations.

• How to solve a system of equations by elimination. Step 1. Write both equations in standard form. If any coefficients are fractions, clear them. Step 2. Make the coefficients of one variable opposites.Decide which variable you will eliminate.Multiply one or both equations so that the coefficients of that variable are opposites. Step 3. Add the equations resulting from Step 2 to eliminate one variable. Step 4. Solve for the remaining variable. Step 5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. Step 6. Write the solution as an ordered pair. Step 7. Check that the ordered pair is a solution to both original equations. $\begin{array}{llccccccccccc} \mathrm{Choose the Most Convenient Method to Solve a System of Linear Equations} \\ \begin{array}{lcclccl} \underset{\mathrm{---------------}}{\mathrm{Graphing}} & & & \underset{\mathrm{---------------------}}{\mathrm{Substitution}} & & & \underset{\mathrm{---------------------}}{\mathrm{Elimination}} \\ \begin{array}{l} \mathrm{Use when you need a} \\ \mathrm{picture of the situation.} \\ \end{array} & & & \begin{array}{l} \mathrm{Use when one equation is} \\ \mathrm{already solved or can be} \\ \mathrm{easily solved for one} \\ \mathrm{variable.} \\ \end{array} & & & \begin{array}{l} \mathrm{Use when the equations are} \\ \mathrm{in standard form.} \\ \end{array} \\ \end{array} \\ \end{array}$

• How To Solve Applications with Systems of Equations Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose variables to represent those quantities. Step 4. Translate into a system of equations. Step 5. Solve the system of equations using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Cost function: The cost function is the cost to manufacture each unit times x, the number of units manufactured, plus the fixed costs.

• Revenue: The revenue function is the selling price of each unit times x, the number of units sold.

• Break-even point: The break-even point is when the revenue equals the costs.

• Linear Equation in Three Variables: A linear equation with three variables, where a, b, c, and d are real numbers and a, b, and c are not all 0, is of the form

Every solution to the equation is an ordered triple, $(x,y,z)$ that makes the equation true.

• How to solve a system of linear equations with three variables. Step 1. Write the equations in standard form If any coefficients are fractions, clear them. Step 2. Eliminate the same variable from two equations. Decide which variable you will eliminate. Work with a pair of equations to eliminate the chosen variable. Multiply one or both equations so that the coefficients of that variable are opposites. Add the equations resulting from Step 2 to eliminate one variable Step 3. Repeat Step 2 using two other equations and eliminate the same variable as in Step 2. Step 4. The two new equations form a system of two equations with two variables. Solve this system. Step 5. Use the values of the two variables found in Step 4 to find the third variable. Step 6. Write the solution as an ordered triple. Step 7. Check that the ordered triple is a solution to all three original equations.

• Matrix: A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with m rows and n columns has order $m\,\times\,n.$ The matrix on the left below has 2 rows and 3 columns and so it has order $2\,\times\,3.$ We say it is a 2 by 3 matrix. Each number in the matrix is called an element or entry in the matrix.

• Row Operations: In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix. Interchange any two rows Multiply a row by any real number except 0 Add a nonzero multiple of one row to another row

• Row-Echelon Form: For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

• How to solve a system of equations using matrices. Step 1. Write the augmented matrix for the system of equations. Step 2. Using row operations get the entry in row 1, column 1 to be 1. Step 3. Using row operations, get zeros in column 1 below the 1. Step 4. Using row operations, get the entry in row 2, column 2 to be 1. Step 5. Continue the process until the matrix is in row-echelon form. Step 6. Write the corresponding system of equations. Step 7. Use substitution to find the remaining variables. Step 8. Write the solution as an ordered pair or triple. Step 9. Check that the solution makes the original equations true.

• Determinant: The determinant of any square matrix $[\begin{array}{l} a\,b \\ c\,d \\ \end{array}],$ where a, b, c, and d are real numbers, is

• Expanding by Minors along the First Row to Evaluate a 3 × 3 Determinant: To evaluate a $3\,\times\,3$ determinant by expanding by minors along the first row, the following pattern:

• Sign Pattern: When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.

• Cramer’s Rule: For the system of equations $\mathrm{\{}\begin{array}{l} {a}_{1}x+{b}_{1}y={k}_{1} \\ {a}_{2}x+{b}_{2}y={k}_{2} \\ \end{array},$ the solution $(x,y)$ can be determined by Notice that to form the determinant D, we use take the coefficients of the variables.

• How to solve a system of two equations using Cramer’s rule. Step 1. Evaluate the determinant D, using the coefficients of the variables. Step 2. Evaluate the determinant ${D}_{x}.$ Use the constants in place of the x coefficients. Step 3. Evaluate the determinant ${D}_{y}.$ Use the constants in place of the y coefficients. Step 4. Find x and y. $x=\frac{{D}_{x}}{D},$$y=\frac{{D}_{y}}{D}.$ Step 5. Write the solution as an ordered pair. Step 6. Check that the ordered pair is a solution to both original equations. Step 7. Dependent and Inconsistent Systems of Equations: For any system of equations, where the value of the determinant $D=0,$$\begin{array}{lcclccl} \mathrm{Value of determinants} & & & \mathrm{Type of system} & & & \mathrm{Solution} \\ D=0\,\mathrm{and}\,{D}_{x},{D}_{y}\,\mathrm{and}\,{D}_{z}\,\mathrm{are all zero} & & & \mathrm{consistent and dependent} & & & \mathrm{infinitely many solutions} \\ D=0\,\mathrm{and}\,{D}_{x},{D}_{y}\,\mathrm{and}\,{D}_{z}\,\mathrm{are not all zero} & & & \mathrm{inconsistent} & & & \mathrm{no solution} \\ \end{array}$ Step 8. Test for Collinear Points: Three points $({x}_{1},{y}_{1}),$$({x}_{2},{y}_{2}),$ and $({x}_{3},{y}_{3})$ are collinear if and only if

• Solutions of a System of Linear Inequalities: Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the x, y coordinate system that includes all the points whose ordered pairs make the inequalities true.

• How to solve a system of linear inequalities by graphing. Step 1. Graph the first inequality.Graph the boundary line.Shade in the side of the boundary line where the inequality is true. Step 2. On the same grid, graph the second inequality.Graph the boundary line.Shade in the side of that boundary line where the inequality is true. Step 3. The solution is the region where the shading overlaps. Step 4. Check by choosing a test point.

##

Review Exercises

Determine Whether an Ordered Pair is a Solution of a System of Equations.

In the following exercises, determine if the following points are solutions to the given system of equations.

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

In the following exercises, without graphing determine the number of solutions and then classify the system of equations.

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

Solve Direct Translation Applications

In the following exercises, translate to a system of equations and solve.

Solve Geometry Applications

In the following exercises, translate to a system of equations and solve.

Solve Uniform Motion Applications

In the following exercises, translate to a system of equations and solve.

Solve Mixture Applications with Systems of Equations

For the following exercises, translate to a system of equations and solve.

Solve Interest Applications

For the following exercises, translate to a system of equations and solve.

Solve Systems of Equations with Three Variables

In the following exercises, determine whether the ordered triple is a solution to the system.

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

Solve Applications using Systems of Linear Equations with Three Variables

Write the Augmented Matrix for a System of Equations.

Write each system of linear equations as an augmented matrix.

Write the system of equations that that corresponds to the augmented matrix.

In the following exercises, perform the indicated operations on the augmented matrices.

Solve Systems of Equations Using Matrices

In the following exercises, solve each system of equations using a matrix.

Evaluate the Determinant of a 2 × 2 Matrix

In the following exercise, evaluate the determinate of the square matrix.

Evaluate the Determinant of a 3 × 3 Matrix

In the following exercise, find and then evaluate the indicated minors.

In the following exercise, evaluate each determinant by expanding by minors along the first row.

In the following exercise, evaluate each determinant by expanding by minors.

Use Cramer’s Rule to Solve Systems of Equations

In the following exercises, solve each system of equations using Cramer’s rule

Solve Applications Using Determinants

In the following exercises, determine whether the given points are collinear.

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

In the following exercises, determine whether each ordered pair is a solution to the system.

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

##

In the following exercises, solve the following systems by graphing.

In the following exercises, solve each system of equations. Use either substitution or elimination.

Solve the system of equations using a matrix.

Solve using Cramer’s rule.

In the following exercises, translate to a system of equations and solve.

1. $\mathrm{\{}\begin{array}{l} x+y=8 \\ y=x-4 \\ \end{array}$ⓐ $(6,2)$ⓑ $(9,-1)$

2. $\mathrm{\{}\begin{array}{l} x+4y=-1 \\ x=3 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} -x+2y=4 \\ y=\frac{1}{2}x-3 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} 3x+2y=6 \\ y=-3x+4 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 3x-2y=2 \\ y=\frac{1}{2}x+3 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} y=-2x+7 \\ y=\frac{2}{3}x-1 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} y=-\frac{1}{3}x+2 \\ x+3y=6 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 3x-8y=20 \\ x+3y=1 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} \frac{1}{3}x-\frac{1}{2}y=1 \\ \frac{3}{4}x-y=\frac{5}{2} \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} 6x-5y=27 \\ 3x+10y=-24 \\ \end{array}$

2. Mollie wants to plant 200 bulbs in her garden, all irises and tulips. She wants to plant three times as many tulips as irises. How many irises and how many tulips should she plant?

3. Leroy spent 20 minutes jogging and 40 minutes cycling and burned 600 calories. The next day, Leroy swapped times, doing 40 minutes of jogging and 20 minutes of cycling and burned the same number of calories. How many calories were burned for each minute of jogging and how many for each minute of cycling?

1. The difference of two supplementary angles is 58 degrees. Find the measures of the angles.

2. The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

3. The perimeter of a city rectangular park is 1428 feet. The length is 78 feet more than twice the width. Find the length and width of the park.

1. Bob left home, riding his bike at a rate of 10 miles per hour to go to the lake. Cheryl, his wife, left 45 minutes $(\frac{3}{4}$ hour) later, driving her car at a rate of 25 miles per hour. How long will it take Cheryl to catch up to Bob?

2. A passenger jet can fly 804 miles in 2 hours with a tailwind but only 776 miles in 2 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.

3. Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?

1. A scientist needs 70 liters of a 40% solution of alcohol. He has a 30% and a 60% solution available. How many liters of the 30% and how many liters of the 60% solutions should he mix to make the 40% solution?

2. When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5$1,585. What is the amount of each loan?

3. $\mathrm{\{}\begin{array}{l} y=\frac{2}{3}x-2 \\ x+3y-z=15 \\ x-3y+z=-2 \\ \end{array}$ⓐ $(-6,5,\frac{1}{2})$ⓑ $(5,\frac{4}{3},-3)$

1. $\mathrm{\{}\begin{array}{l} x+\frac{5}{2}y+z=-2 \\ 2x+2y+\frac{1}{2}z=-4 \\ \frac{1}{3}x-y-z=1 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 2x+3y+z=12 \\ x+y+z=9 \\ 3x+4y+2z=20 \\ \end{array}$

3. After attending a major league baseball game, the patrons often purchase souvenirs. If a family purchases 4 t-shirts, a cap and 1 stuffed animal their total is $135. A couple buys 2 t-shirts, a cap and 3 stuffed animals for their nieces and spends$115. Another couple buys 2 t-shirts, a cap and 1 stuffed animal and their total is $85. What is the cost of each item?

1. $\mathrm{\{}\begin{array}{l} 4x+3y=-2 \\ x-2y-3z=7 \\ 2x-y+2z=-6 \\ \end{array}$

2. $[\begin{array}{rccrccr} 1 & & & 0 & & & -3 \\ 1 & & & -2 & & & 0 \\ 0 & & & -1 & & & 2 \\ \end{array}\,|\,\begin{array}{r} -1 \\ -2 \\ 3 \\ \end{array}]$

3. $[\begin{array}{lccrccr} 1 & & & -3 & & & -2 \\ 2 & & & 2 & & & -1 \\ 4 & & & -2 & & & -3 \\ \end{array}\,|\,\begin{array}{r} 4 \\ -3 \\ -1 \\ \end{array}]$

ⓐ Interchange rows 2 and 3. ⓑ Multiply row 1 by 2. ⓒ Multiply row 3 by $-2$ and add to row 2.

1. $\mathrm{\{}\begin{array}{l} 2x-y+3z=-3 \\ -x+2y-z=10 \\ x+y+z=5 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} x+2y-3z=-1 \\ x-3y+z=1 \\ 2x-y-2z=2 \\ \end{array}$

3. $[\begin{array}{lccr} 8 & & & -4 \\ 5 & & & -3 \\ \end{array}]$

1. $\mathrm{\{}\begin{array}{l} x-3y=-9 \\ 2x+5y=4 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 2x+5y=4 \\ 3y-z=3 \\ 4x+3z=-3 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 3x+4y-3z=-2 \\ 2x+3y-z=-12 \\ x+y-2z=6 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} 4x+y>6 \\ 3x-y\le{}12 \\ \end{array}$

ⓐ $(2,-1)$ⓑ $(3,-2)$

2. $\mathrm{\{}\begin{array}{l} y\le{}3x+2 \\ y>-x-1 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 2x-3y<6 \\ 3x+4y\ge{}12 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} x+3y<5 \\ y\ge-\frac{1}{3}x+6 \\ \end{array}$

2. Roxana makes bracelets and necklaces and sells them at the farmers’ market. She sells the bracelets for $12 each and the necklaces for$18 each. At the market next weekend she will have room to display no more than 40 pieces, and she needs to sell at least $500 worth in order to earn a profit.

ⓐ Write a system of inequalities to model this situation. ⓑ Graph the system. ⓒ Should she display 26 bracelets and 14 necklaces? ⓓ Should she display 39 bracelets and 1 necklace?

3. $\mathrm{\{}\begin{array}{l} x-y=5 \\ x+2y=-4 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} x+4y=6 \\ -2x+y=-3 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} x+y-z=-1 \\ 2x-y+2z=8 \\ -3x+2y+z=-9 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} -3x+y+z=-4 \\ -x+2y-2z=1 \\ 2x-y-z=-1 \\ \end{array}$

1. Evaluate the determinant by expanding by minors:$|\begin{array}{rccrccr} 3 & & & -2 & & & -2 \\ 2 & & & -1 & & & 4 \\ -1 & & & 0 & & & -3 \\ \end{array}|.$

2. A pharmacist needs 20 liters of a 2% saline solution. He has a 1% and a 5% solution available. How many liters of the 1% and how many liters of the 5% solutions should she mix to make the 2% solution?

3. The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of $65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of$140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 can of popcorn for a total sales of $250. What is the cost of each item?

1. Translate to a system of inequalities and solve.

Andi wants to spend no more than $50 on Halloween treats. She wants to buy candy bars that cost$1 each and lollipops that cost $0.50 each, and she wants the number of lollipops to be at least three times the number of candy bars.

ⓐ Write a system of inequalities to model this situation. ⓑ Graph the system. ⓒ Can she buy 20 candy bars and 40 lollipops?