➕ Algebra 1 · Equations
Equations with Variables on Both Sides: true or false
Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.
All statements and answers
The equation x + 4 = x + 9 has no solution.
✓ True. Subtract x from both sides to get 4 = 9, which is false for every x.
The equation 2(x + 3) = 2x + 6 is true for every number x.
✓ True. Distributing makes the left side 2x + 6, the same expression as the right side.
If you get 0 = 0 while solving, the answer is x = 0.
✗ False. 0 = 0 is true no matter what x is, so every number is a solution. The value x = 0 is only one of them.
If the variables cancel and you are left with 3 = 8, you must have made an arithmetic mistake.
✗ False. Not necessarily. Check your work, but if your steps are right, a false leftover like 3 = 8 means the equation has no solution.
You can solve 7x + 2 = 3x + 14 by subtracting 3x first or by subtracting 7x first, and both routes give x = 3.
✓ True. Subtract 3x: 4x + 2 = 14, so x = 3. Subtract 7x: 2 = −4x + 14, so −12 = −4x and x = 3.
Every equation with variables on both sides has exactly one solution.
✗ False. Not always. x + 1 = x + 2 has no solution, and 2x = 2x has infinitely many.
The solution of 20 − x = 3x − 16 is x = 2.
✗ False. Add x to both sides: 20 = 4x − 16. Add 16: 36 = 4x, so x = 9. Check x = 2: the left side is 18 and the right side is −10.
6x + 18 = 6(x + 3) has no solution, because the x-terms cancel.
✗ False. The x-terms do cancel, but the leftover is 18 = 18, which is true. So every number is a solution. Always read what is left.
The solution of x/2 + 4 = x/6 is x = −12.
✓ True. Multiply every term by 6: 3x + 24 = x. Subtract x: 2x + 24 = 0, so x = −12. Check: −6 + 4 = −2 and −12/6 = −2.
An equation with variables on both sides can have a fraction as its solution. For example, 4x + 1 = x + 3 gives x = 2/3.
✓ True. Subtract x: 3x + 1 = 3. Subtract 1: 3x = 2. Divide by 3: x = 2/3. Check: both sides equal 11/3.