Systems of Linear Equations Practice

What this problem type covers

Each problem gives two linear equations in x and y with exactly one solution. Coefficients stay small at first and grow later, so the arithmetic gets heavier while the method stays the same.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   Solve the system for x and y.
   −3x − 3y = 3
   2x + y = 1
   • ①x = −3, y = 2
   • ②x = 2, y = −3
   • ③x = −2, y = 3
   • ④x = 2, y = 3
   • ⑤x = −2, y = −3
   Show solution

   ②x = 2, y = −3

   1. Scale the two equations so that y cancels when you add them.
   2. That leaves a single equation, giving x = 2.
   3. Substitute back into either equation to get y = −3.
   4. Check both equations with x = 2, y = −3 — both hold.
2. 2 · ★2
   Solve the system for x and y.
   2x + 3y = −3
   −3x + 2y = −2
   • ①x = 0, y = 0
   • ②x = −1, y = 0
   • ③x = 1, y = −1
   • ④x = 0, y = −1
   • ⑤x = 0, y = 1
   Show solution

   ④x = 0, y = −1

   1. Scale the two equations so that y cancels when you add them.
   2. That leaves a single equation, giving x = 0.
   3. Substitute back into either equation to get y = −1.
   4. Check both equations with x = 0, y = −1 — both hold.
3. 3 · ★3
   Solve the system for x and y.
   x + y = 1
   5x − 2y = 47
   • ①x = −7, y = −6
   • ②x = −7, y = 6
   • ③x = 7, y = −6
   • ④x = 7, y = 6
   • ⑤x = −6, y = 7
   Show solution

   ③x = 7, y = −6

   1. Scale the two equations so that y cancels when you add them.
   2. That leaves a single equation, giving x = 7.
   3. Substitute back into either equation to get y = −6.
   4. Check both equations with x = 7, y = −6 — both hold.
4. 4 · ★4
   Solve the system for x and y.
   x − y = −4
   5x − 6y = −28
   • ①x = −4, y = 8
   • ②x = −4, y = −8
   • ③x = 4, y = −8
   • ④x = 4, y = 8
   • ⑤x = 8, y = 4
   Show solution

   ④x = 4, y = 8

   1. Scale the two equations so that y cancels when you add them.
   2. That leaves a single equation, giving x = 4.
   3. Substitute back into either equation to get y = 8.
   4. Check both equations with x = 4, y = 8 — both hold.
5. 5 · ★5
   Solve the system for x and y.
   2x − 2y = 2
   x + 7y = −23
   • ①x = 2, y = 3
   • ②x = −2, y = 3
   • ③x = −2, y = −3
   • ④x = 2, y = −3
   • ⑤x = −3, y = −2
   Show solution

   ③x = −2, y = −3

   1. Scale the two equations so that y cancels when you add them.
   2. That leaves a single equation, giving x = −2.
   3. Substitute back into either equation to get y = −3.
   4. Check both equations with x = −2, y = −3 — both hold.

Where people usually go wrong

• Adding the equations without scaling first. Elimination only works when one variable has matching magnitudes.
• Forgetting to multiply the right-hand side. Scaling an equation means scaling every term, constants included.
• Sign errors when subtracting. Subtracting a negative coefficient adds — this is where most wrong answers come from.
• Solving for one variable and stopping. The answer is a pair; half an answer scores nothing.
• Never checking. Substituting back into both equations takes ten seconds and catches nearly every arithmetic slip.