RANGER Math practice · Algebra · ages 14+
Free systems of equations practice
A system of equations is two equations that share the same x and y. Ranger's Math Trail practices solving them by elimination: first simple ones where adding the equations cancels y, then ones where you multiply first to make a variable cancel.
Type the answer as x, y (like 3, −2). The worked answer shows each equation being multiplied, the subtraction that cancels a variable, and a check in both equations.
Practice systems of equations now
Pick a level
Free, no sign-up. The game asks your age once, then every answer comes with a hint and a step-by-step Show me how.
How to do it
- Line up the two equations: x terms, y terms, numbers.
- Multiply one or both equations so one variable has the same number in front.
- Add or subtract the equations to cancel that variable, then solve for the other one.
- Put that value back into either equation to find the second variable, then check both equations.
Worked examples
Example 1Solve the system:
x + y = 1
x − y = 5
- Add the two equations: the y terms cancel. (x + y) + (x − y) = 1 + 5, so 2x = 6.
- x = 6 ÷ 2 = 3.
- Put x back into x + y = 1: 3 + y = 1, so y = −2.
- Check the second one: 3 − (−2) = 5. ✔
Answer: (3, −2)
Try one like it →Example 2Solve the system:
4x + 2y = −36
3x − y = −12
- Make the x terms match: multiply the first equation by 3 and the second by 4.
- Subtract to cancel x: (6 − (−4))y = −108 − (−48), so 10y = −60.
- y = −60 ÷ 10 = −6.
- Put y = −6 into the first equation: 4x + (−12) = −36, so 4x = −24 and x = −6.
- Check both: −36 = −36 and −12 = −12. ✔
Answer: (−6, −6)
Try one like it →Mistakes to watch for
- If the y terms are already opposites (+y and −y), just add the equations.
- When you multiply an equation, multiply every term, including the number on the right side.
The game
Questions
How do I enter the answer?
Type x, then a comma, then y: 3, −2. The number pad has a comma key.
What age is it for?
Systems of equations come in around grade 8 and Algebra 1 (ages 13 to 15).
More math practice: Addition · Subtraction · Multiplication facts · Division and long division · Place value and rounding · Fractions · Decimals · Percents and ratios · Negative numbers · Order of operations · Exponents and square roots · Solving equations · Slope and linear equations · Factoring quadratics · Area, perimeter and volume · Angles · Pythagorean theorem · Circles · Mean, median, mode and range · All math topics