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MPM2D Linear Systems
Grade 10 Unit 1 — practice questions with full solutions

Solving two equations at once by graphing, substitution and elimination — plus the word problems that always appear on tests.

MPM2D · Grade 10 Analytic Geometry 6 core 1 challenge
1Core
-2-11234567-4-3-2-1123456xy(2, 1)y = 2x - 3y = -x + 3
Use the graph above to solve the linear system, then verify your answer algebraically in both equations.
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  1. The solution to a system is the point where the lines cross.
  2. Reading the graph, they intersect at (2, 1).
  3. Check in y = 2x − 3: 2(2) − 3 = 1. ✔
  4. Check in y = −x + 3: −(2) + 3 = 1. ✔
  5. Both are satisfied, so (2, 1) is confirmed.
Final answer(2, 1)
2Core
Solve by substitution:   y = 3x − 4   and   2x + y = 11
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  1. The first equation is already solved for y, so substitute it into the second.
  2. 2x + (3x − 4) = 11.
  3. Simplify: 5x − 4 = 11, so 5x = 15 and x = 3.
  4. Substitute back: y = 3(3) − 4 = 5.
Final answer(3, 5)
3Core
Solve by elimination:   3x + 2y = 16   and   5x − 2y = 8
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  1. The y terms are already opposites, so add the two equations directly.
  2. (3x + 5x) + (2y − 2y) = 16 + 8 gives 8x = 24.
  3. So x = 3.
  4. Substitute into the first: 3(3) + 2y = 16, so 2y = 7 and y = 3.5.
Final answer(3, 3.5)
4Core
Solve by elimination:   4x + 3y = 10   and   3x + 5y = 13
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  1. Neither variable eliminates yet, so multiply to match coefficients. Multiply the first by 5 and the second by 3.
  2. 20x + 15y = 50 and 9x + 15y = 39.
  3. Subtract: 11x = 11, so x = 1.
  4. Substitute into 4(1) + 3y = 10: 3y = 6, so y = 2.
Final answer(1, 2)
5Core
Adult tickets cost $12 and student tickets cost $7. A total of 250 tickets were sold for $2400. How many of each were sold?
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  1. Let a be adult tickets and s student tickets.
  2. Ticket count: a + s = 250. Revenue: 12a + 7s = 2400.
  3. From the first, s = 250 − a. Substitute: 12a + 7(250 − a) = 2400.
  4. Expand: 12a + 1750 − 7a = 2400, so 5a = 650 and a = 130.
  5. Then s = 120. Check: 12(130) + 7(120) = 1560 + 840 = 2400. ✔
Final answer130 adult, 120 student
6Core
A canoe travels 24 km downstream in 2 hours and the same 24 km upstream in 3 hours. Find the speed of the canoe in still water and the speed of the current.
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  1. Let c be the canoe speed and r the current speed.
  2. Downstream the speeds add: c + r = 24 ÷ 2 = 12.
  3. Upstream they subtract: c − r = 24 ÷ 3 = 8.
  4. Add the two equations: 2c = 20, so c = 10.
  5. Then r = 12 − 10 = 2.
Final answercanoe 10 km/h, current 2 km/h
7Challenge
For what value of k does the system   2x + 3y = 7   and   6x + ky = 21   have infinitely many solutions?
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  1. Infinitely many solutions means the two equations describe the same line.
  2. Compare the x terms: 6 ÷ 2 = 3, so the second equation must be exactly 3 times the first.
  3. Check the constants: 3 × 7 = 21. ✔ This is consistent.
  4. So the y term must satisfy k = 3 × 3 = 9.
  5. Note that if the constant had not matched, no value of k would work — the lines would be parallel instead.
Final answerk = 9

MPM2D Linear Systems — common questions

Short answers to the things students ask most about this unit.

Should I use substitution or elimination?

Use substitution when one equation is already solved for a variable, or easily can be. Use elimination when the equations are both in standard form and coefficients line up.

What does it mean if a system has no solution?

The lines are parallel — same slope, different intercept — so they never cross. If both equations describe the same line, there are infinitely many solutions instead.

More MPM2D practice — grade 10 academic

MPM2D Analytic GeometryMPM2D Factoring QuadraticsMPM2D Quadratic Relations & GraphingMPM2D Solving Quadratic EquationsAll MPM2D units
← All assignmentsUnit 2: Analytic Geometry →
Practice is step one

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