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

Solving an Equation with Variables on Both Sides

Solving 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.

Optional Video: Solve an Equation with Variable Terms on Both Sides

Example

Solve: \(5x=4x+7.\)

Solution

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

 Solving an Equation with Variables on Both Sides
We don’t want any variables on the right, so subtract the \(4x\).Solving an Equation with Variables on Both Sides
Simplify.Solving an Equation with Variables on Both Sides
We have all the variables on one side and the constants on the other. We have solved the equation. 
Check:Solving an Equation with Variables on Both Sides 
Substitute 7 for \(x\).Solving an Equation with Variables on Both Sides 
 Solving an Equation with Variables on Both Sides 
 Solving an Equation with Variables on Both Sides 

Example

Solve: \(5y-8=7y.\)

Solution

The only constant, \(-8,\) is on the left side of the equation and variable, \(y,\) is on both sides. Let’s leave the constant on the left and collect the variables to the right.

 Solving an Equation with Variables on Both Sides
Subtract \(5y\) from both sides.Solving an Equation with Variables on Both Sides
Simplify.Solving an Equation with Variables on Both Sides
We have the variables on the right and the constants on the left. Divide both sides by 2.Solving an Equation with Variables on Both Sides
Simplify.Solving an Equation with Variables on Both Sides
Rewrite with the variable on the left.Solving an Equation with Variables on Both Sides
Check: Let \(y=-4\). 
Solving an Equation with Variables on Both Sides 
Solving an Equation with Variables on Both Sides 
Solving an Equation with Variables on Both Sides 
Solving an Equation with Variables on Both Sides 

Example

Solve: \(7x=-x+24.\)

Solution

The only constant, \(24,\) is on the right, so let the left side be the variable side.

 Solving an Equation with Variables on Both Sides
Remove the \(-x\) from the right side by adding \(x\) to both sides.Solving an Equation with Variables on Both Sides
Simplify.Solving an Equation with Variables on Both Sides
All the variables are on the left and the constants are on the right. Divide both sides by 8.Solving an Equation with Variables on Both Sides
Simplify.Solving an Equation with Variables on Both Sides
Check: Substitute \(x=3\). 
Solving an Equation with Variables on Both Sides 

Optional Video: Solving Multi-Step Equations

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