Mathematics » Solving Linear Equations I » Solve Equations with Variables and Constants on Both Sides

Solving an Equation with Constants on Both Sides

Solving 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.

Example

Solve: \(4x+6=-14.\)

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.

 Solving an Equation with Constants on Both Sides
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.Solving an Equation with Constants on Both Sides
Simplify.Solving an Equation with Constants on Both Sides
Now all the \(x\)s are on the left and the constant on the right. 
Use the Division Property of Equality.Solving an Equation with Constants on Both Sides
Simplify.Solving an Equation with Constants on Both Sides
Check:Solving an Equation with Constants on Both Sides 
Let \(x=-5\).Solving an Equation with Constants on Both Sides 
 Solving an Equation with Constants on Both Sides 
 Solving an Equation with Constants on Both Sides 

Example

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.

 Solving an Equation with Constants on Both Sides
Add 7 to both sides.Solving an Equation with Constants on Both Sides
Simplify.Solving an Equation with Constants on Both Sides
The variables are now on one side and the constants on the other. 
Divide both sides by 2.Solving an Equation with Constants on Both Sides
Simplify.Solving an Equation with Constants on Both Sides
Check:Solving an Equation with Constants on Both Sides 
Substitute: \(y=11\).Solving an Equation with Constants on Both Sides 
 Solving an Equation with Constants on Both Sides 
 Solving an Equation with Constants on Both Sides 

Optional Video: Solving Multi-Step Equations

[Attributions and Licenses]


This is a lesson from the tutorial, Solving Linear Equations I and you are encouraged to log in or register, so that you can track your progress.

Log In

Discussion


Do NOT follow this link or you will be banned from the site!