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.

## 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.  We don’t want any variables on the right, so subtract the $$4x$$.  Simplify.  We have all the variables on one side and the constants on the other. We have solved the equation. Check:  Substitute 7 for $$x$$.      ## 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.  Subtract $$5y$$ from both sides.  Simplify.  We have the variables on the right and the constants on the left. Divide both sides by 2.  Simplify.  Rewrite with the variable on the left.  Check: Let $$y=-4$$.        ## Example

Solve: $$7x=-x+24.$$

### Solution

The only constant, $$24,$$ is on the right, so let the left side be the variable side.  Remove the $$-x$$ from the right side by adding $$x$$ to both sides.  Simplify.  All the variables are on the left and the constants are on the right. Divide both sides by 8.  Simplify.  Check: Substitute $$x=3$$.  