8.3 Solve equations with variables and constants on both sides

Solve an equation with constants on both sides

You may have noticed that 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’ll see how to solve equations where the variable terms and/or constant terms 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, step by step, 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 started 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.

Solve: $4x+6=\mathrm{-14}.$

Solution

In this equation, the variable is 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’ll write the labels above the equation to help us remember what goes where.

Since the left side is the variable side, the 6 is out of place. We must "undo" adding 6 by subtracting 6, and to keep the equality we must subtract 6 from both sides. Use the Subtraction Property of Equality.
Simplify.
Now all the $x$ s are on the left and the constant on the right.
Use the Division Property of Equality.
Simplify.
Check:
Let $x=\mathrm{-5}$ .

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Solve: $2y-7=15.$

Solution

Notice that the variable is only on the left side of the equation, so this will be the variable side and the right side will be the constant side. Since the left side is the variable side, the $7$ is out of place. It is subtracted from the $2y,$ so to ‘undo’ subtraction, add $7$ to both sides.

Add 7 to both sides.
Simplify.
The variables are now on one side and the constants on the other.
Divide both sides by 2.
Simplify.
Check:
Substitute: $y=11$ .

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Solve an equation with variables on both sides

What if there are variables on both sides of the equation? We will start like we did above—choosing 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. Remember, what you do to the left side of the equation, you must do to the right side too.