Equations with Variables on Both Sides
Free printable worksheets on solving equations with variables on both sides, like 5x - 3 = 2x + 9, with negative numbers. Answer key included.
Each equation has a variable term on both sides, like 5x - 3 = 2x + 9. Start by collecting the variable terms on one side: subtracting 2x from both sides leaves 3x - 3 = 9, which is a two-step equation. Add 3 to get 3x = 12, then divide by 3, so x = 4.
The sheet uses negative numbers, as eighth grade does. Solutions, constants and coefficients can all be below zero, as in -3x + 4 = 2x - 11, where x = 3. About half the solutions on a sheet are negative. Every equation has exactly one solution, a whole number, and the answer key gives it. Turn off Negative Numbers under Advanced options for a sheet where nothing goes negative at any step.
What this worksheet covers
- Solve equations with the variable on both sides
- Collect the variable terms on one side of an equation
- Solve equations with negative coefficients and solutions
Common Core standards
- 8.EE.C.7 Solve linear equations in one variable.
Frequently asked questions
How do you solve an equation with variables on both sides?
Get the variable terms together first. In 7x + 2 = 3x + 18, subtract 3x from both sides to get 4x + 2 = 18, subtract 2 to get 4x = 16, and divide by 4, so x = 4. Check by putting 4 back in: 7 × 4 + 2 = 30 and 3 × 4 + 18 = 30.
Which side should the variable go on?
Either works, but subtracting the smaller variable term keeps the coefficient positive. In 2x + 9 = 5x - 3, subtracting 2x gives 9 = 3x - 3, so 3x = 12 and x = 4. Subtracting 5x instead gives -3x + 9 = -3: the same answer, but with a negative number to divide by.
What if the variable cancels out?
Then the equation has no solution or infinitely many. Subtracting 2x from 2x + 3 = 2x + 5 leaves 3 = 5, which is never true, so no value of x works. Subtracting 2x from 2x + 3 = 2x + 3 leaves 3 = 3, which is always true, so every value works. On these sheets the two sides always have different coefficients, so every equation has exactly one solution.
What grade is this?
Eighth grade. It follows two-step equations in seventh grade: once the variable terms are collected, what is left is a two-step equation.