2.3 Solve Equations with Variables and Constants on Both Sides

Learning objectives

Learning objectives

• Solve an equation with constants on both sides

• Solve an equation with variables on both sides

• Solve an equation with variables and constants on both sides

Solve Equations with Constants on Both Sides

In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both are on both sides of the equation.

Our strategy will involve choosing one side of the equation to be the “variable side”, and the other side of the equation to be the “constant side.” Then, we will use the Subtraction and Addition Properties of Equality to get all the variable terms together on one side of the equation and the constant terms together on the other side.

By doing this, we will transform the equation that began with variables and constants on both sides into the form $ax=b.$ We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.

Example 2.27

Solution

In this equation, the variable is found only on the left side. It makes sense to call the left side the “variable” side. Therefore, the right side will be the “constant” side. We will write the labels above the equation to help us remember what goes where.

Since the left side is the “$x$”, or variable side, the 8 is out of place. We must “undo” adding 8 by subtracting 8, and to keep the equality we must subtract 8 from both sides.

Example 2.28

Solution

Notice, the variable is only on the left side of the equation, so we will call this side the “variable” side, and the right side will be the “constant” side. Since the left side is the “variable” side, the 9 is out of place. It is subtracted from the $8y$, so to “undo” subtraction, add 9 to both sides. Remember, whatever you do to the left, you must do to the right.

Solve Equations with Variables on Both Sides

What if there are variables on both sides of the equation? For equations like this, begin as we did above—choose a “variable” side and a “constant” side, and then use the subtraction and addition properties of equality to collect all variables on one side and all constants on the other side.

Example 2.29

Solution

Here the variable is on both sides, but the constants only appear on the right side, so let’s make the right side the “constant” side. Then the left side will be the “variable” side.

Example 2.30

Solution

The only constant is on the left and the $y$’s are on both sides. Let’s leave the constant on the left and get the variables to the right.

Example 2.31

Solution

The only constant is on the right, so let the left side be the “variable” side.

Solve Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. It may take several steps to solve this equation, so we need a clear and organized strategy.

Example 2.32

We’ll list the steps below so you can easily refer to them. But we’ll call this the ‘Beginning Strategy’ because we’ll be adding some steps later in this chapter.

In Step 1, a helpful approach is to make the “variable” side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier.

Example 2.33

Solution

In the first step, choose the variable side by comparing the coefficients of the variables on each side.

Example 2.34

Solution

In the first step, choose the variable side by comparing the coefficients of the variables on each side.

Since $13>7$, make the right side the “variable” side and the left side the “constant” side.

In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we just follow the steps of our strategy to get the solution!

Example 2.35

Solution

Since $\frac{5}{4}>\frac{1}{4}$, make the left side the “variable” side and the right side the “constant” side.

We will use the same strategy to find the solution for an equation with decimals.

Example 2.36

Solution

Since $7.8>5.4$, make the left side the “variable” side and the right side the “constant” side.

Section 2.3 Exercises

Section 2.3 Exercises

Solve Equations with Constants on Both Sides

In the following exercises, solve the following equations with constants on both sides.

Solve Equations with Variables on Both Sides

In the following exercises, solve the following equations with variables on both sides.

Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the following equations with variables and constants on both sides.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. Solve: $3x+4=-8.$

2. Solve: $5a+3=-37.$

3. Solve: $5y-9=16.$

4. Solve: $3m-8=19.$

1. Solve: $6n=5n-10.$

2. Solve: $-6c=-7c-1.$

3. Solve: $3p-14=5p.$

4. Solve: $8m+9=5m.$

1. Solve: $12j=-4j+32.$

2. Solve: $8h=-4h+12.$

3. Solve: $12x+8=6x+2.$

4. Solve: $9y+4=7y+12.$

1. Solve: $8q-5=-4q+7.$

2. Solve: $7n-3=n+3.$

3. Solve: $2a-2=6a+18.$

4. Solve: $4k-1=7k+17.$

1. Solve: $\frac{7}{8}x-12=-\frac{1}{8}x-2.$

2. Solve: $\frac{7}{6}y+11=\frac{1}{6}y+8.$

3. Solve: $2.8x+12=-1.4x-9.$

4. Solve: $3.6y+8=1.2y-4.$

1. $12x-8=64$

2. $15y+7=97$

3. $3m+9=-15$

4. $-77=9b-5$

1. $60=-21x-24$

2. $-14q-2=16$

3. $21k=20k-11$

4. $8x+27=11x$

1. $b=-4b-15$

2. $5z=39-8z$

3. $4x+\frac{3}{4}=3x$

4. $-11r-8=-7r$

1. $6x-17=5x+2$

2. $21+18f=19f+14$

3. $12q-5=9q-20$

4. $8c+7=-3c-37$

1. $7x-17=-8x+13$

2. $9p+14=6+4p$

3. $3y-4=12-y$

4. $\frac{7}{4}m-7=\frac{3}{4}m-13$

1. $11-\frac{1}{5}a=\frac{4}{5}a+4$

2. $\frac{5}{4}a+15=\frac{3}{4}a-5$

3. $\frac{3}{5}p+2=\frac{4}{5}p-1$

4. $13z+6.45=8z+23.75$

1. $2.7w-80=1.2w+10$

2. $6.6x-18.9=3.4x+54.7$

3. Making a fence Jovani has 150 feet of fencing to make a rectangular garden in his backyard. He wants the length to be 15 feet more than the width. Find the width, w, by solving the equation $150=2w+30+2w$.

4. Solve the equation $10x+14=-2x+38$ explaining all the steps of your solution as in the examples in this section.

1. Is $x=-2$ a solution to the equation $5-2x=-4x+1$ ? How do you know?