10.1 Systems of Two Linear Equations
Key Takeaways
- A solution of a system of two linear equations in two variables is an ordered pair that makes both equations true; on a graph it is the intersection of the two lines.
- Use substitution when one equation is already solved for a variable; use elimination when you can match coefficients and add or subtract so one variable cancels.
- If the variable terms cancel and a false leftover such as 0 = 3 remains, the system has no solution (parallel distinct lines). A true leftover such as 0 = 0 means infinitely many solutions (the same line).
- One purchase with two unknown prices is one line of possible pairs; two independent purchases give two equations and can pin down both prices.
- Desmos graphs the intersection as a check after you write the system; algebra is the setup, especially when an SPR box needs an exact value.
This independent OpenExamPrep chapter finishes the PSAT 8/9 Algebra list that started in 9.1 Linear Equations in One Variable, 9.2 Linear Equations in Two Variables, and 9.3 Linear Functions. College Board's Assessment Framework for the Digital SAT Suite names five Algebra skill/knowledge testing points. This chapter teaches the remaining two—systems of two linear equations in two variables and linear inequalities in one or two variables—and then walks through creating linear models from context, because the Algebra domain also asks you to create linear equations and inequalities from science, social studies, and everyday stories. Algebra is about 42.5% of operational Math on PSAT 8/9, 16–18 questions. About 30% of Math is in context. A calculator is allowed on every Math item; Bluebook's Desmos graphing calculator is a check, not a substitute for writing the two equations. OpenExamPrep does not claim official approval, review, partnership, or exact equivalence with College Board materials.
A system of two linear equations in two variables is a pair of first-power equations that you try to satisfy at the same time. A solution is an ordered pair (x, y) that makes both equations true. Graphically, that pair is where the two lines meet—if they meet. One equation by itself, as in 9.2, is still a whole line of pairs. The second independent equation is what pins the pair down. This test does not assess trigonometry, circle theorems, or rational or radical equations; stay on two linear lines.
| Outcome | Algebra leftover | Graph | Meaning |
|---|---|---|---|
| One solution | A unique pair, such as x = 6, y = 4 | Two lines cross at one point | Different slopes |
| No solution | A false number sentence, such as 0 = 3 | Parallel distinct lines | Same slope, different intercepts |
| Infinitely many | A true number sentence, such as 0 = 0 | The same line twice | Same slope and same intercept |
Keep the language items actually use: one solution, no solution, or infinitely many solutions. You do not need extra vocabulary to earn the point, but comparing slopes is the fastest way to predict the graph before you grind arithmetic.
Substitution
Solve one equation for one variable, then replace that variable in the other equation. Substitution is fastest when a coefficient is 1 or when an equation already says x = ... or y = ....
Example. Solve x + y = 10 and 2x + y = 16.
From the first equation, y = 10 - x. Substitute into the second: 2x + (10 - x) = 16. Combine like terms: x + 10 = 16. Subtract 10: x = 6. Then y = 10 - 6 = 4.
Check both originals: 6 + 4 = 10, and 2(6) + 4 = 12 + 4 = 16. The pair (6, 4) is the solution. Checking only one equation is not enough—an ordered pair can sit on one line and miss the other.
Example. Solve x = y + 3 and 2x + y = 18.
The first equation is already solved for x. Substitute: 2(y + 3) + y = 18. Distribute: 2y + 6 + y = 18. 3y + 6 = 18. 3y = 12. y = 4, so x = 4 + 3 = 7.
Check: 7 = 4 + 3, and 2(7) + 4 = 14 + 4 = 18.
Example. Solve 5x - y = 9 and y = 2x - 3.
Replace y: 5x - (2x - 3) = 9. Distribute the minus: 5x - 2x + 3 = 9. 3x + 3 = 9. 3x = 6. x = 2, so y = 2(2) - 3 = 1.
Check: 5(2) - 1 = 10 - 1 = 9, and 1 = 4 - 3. A frequent error is writing 5x - (2x - 3) as 5x - 2x - 3, which drops the sign on the 3 and yields the wrong pair.
Elimination (addition or subtraction)
Add or subtract the equations so one variable cancels. Multiply one or both equations first if the coefficients do not already match.
Example. Solve 3x + 2y = 16 and x + 2y = 8.
Both have 2y. Subtract the second from the first: (3x + 2y) - (x + 2y) = 16 - 8 2x = 8 x = 4
Plug into x + 2y = 8: 4 + 2y = 8, so 2y = 4, y = 2.
Check: 3(4) + 2(2) = 12 + 4 = 16, and 4 + 2(2) = 8.
Example. Solve 2x + 3y = 13 and 4x - y = 5.
Multiply the first equation by 2 so the x terms match: 4x + 6y = 26. Subtract the second: (4x + 6y) - (4x - y) = 26 - 5. 4x + 6y - 4x + y = 21 7y = 21 y = 3
From 4x - y = 5: 4x - 3 = 5, so 4x = 8, x = 2.
Check: 2(2) + 3(3) = 4 + 9 = 13, and 4(2) - 3 = 8 - 3 = 5. Watch the minus when you subtract (4x - y): subtracting a subtracted y adds y. If you write 6y - y = 5y, you will get y = 21/5 and fail the check.
Graphing: the intersection
Each equation is a line. The solution of the system is the intersection point.
Example. Solve y = x + 2 and y = -x + 8.
Set the right sides equal: x + 2 = -x + 8. 2x = 6 x = 3, so y = 3 + 2 = 5.
Check the second line: -3 + 8 = 5. Intersection (3, 5).
A table shows the same meeting point. The y-values match only at x = 3:
| x | y = x + 2 | y = -x + 8 |
|---|---|---|
| 0 | 2 | 8 |
| 1 | 3 | 7 |
| 2 | 4 | 6 |
| 3 | 5 | 5 |
| 4 | 6 | 4 |
On Desmos, type both equations and tap the intersection. That is a strong check when coefficients are messy. Algebra is usually faster when numbers are small integers, and it is the safer path for an exact fraction on a student-produced response (SPR). Desmos will happily confirm a mistyped equation, so write the system on the school scratch paper first (write your name; do not tear the packet). Calculator policy lives in 3.1 Devices, What to Bring, and Calculator Policy. SPR boxes live in 15.1 Student-Produced Response Questions.
No solution: parallel distinct lines
Example. Consider 2x + y = 5 and 2x + y = 8.
Subtract: 0 = -3, which is false. No solution.
Slope-intercept form: y = -2x + 5 and y = -2x + 8. Same slope -2, different intercepts 5 and 8. Parallel lines never meet. If you graph both in Desmos, you see two rails that never cross. Do not invent a fake intersection by averaging 5 and 8, and do not report x = 0 just because the x terms cancelled.
If two slopes match and the intercepts also match, you are not in this case—you have the same line, next heading.
Infinitely many solutions: the same line
Example. Consider x + y = 6 and 2x + 2y = 12.
Divide the second equation by 2 and you get x + y = 6 again. Subtract: 0 = 0, a true sentence with no leftover x or y. Infinitely many solutions—every pair on the line x + y = 6 works, such as (0, 6), (1, 5), and (6, 0).
Example. Consider y = 3x - 1 and 6x - 2y = 2.
Solve the second for y: -2y = -6x + 2, so y = 3x - 1. The equations are identical. One line, infinitely many points. In Desmos the two graphs stack and can look like a single line, which is the clue.
Word problems: two items, two prices
A school store sells notebooks for n dollars each and pens for p dollars each. Four notebooks and three pens cost $25. Two notebooks and five pens cost $23.
4n + 3p = 25 2n + 5p = 23
Multiply the second equation by 2: 4n + 10p = 46. Subtract the first: (4n + 10p) - (4n + 3p) = 46 - 25 7p = 21 p = 3
From 2n + 5(3) = 23: 2n + 15 = 23, so 2n = 8, n = 4.
Check: 4(4) + 3(3) = 16 + 9 = 25, and 2(4) + 5(3) = 8 + 15 = 23. Notebooks are $4; pens are $3.
The trap is treating one purchase as enough. 4n + 3p = 25 by itself still allows many pairs, such as n = 4, p = 3 and also n = 1, p = 7 (because 4 + 21 = 25). You need two independent purchases to pin down both prices. That is why this skill is separate from 9.2.
Second story. Three adult tickets and two student tickets cost $39. One adult ticket and four student tickets cost $33.
3a + 2s = 39 a + 4s = 33, so a = 33 - 4s
3(33 - 4s) + 2s = 39 99 - 12s + 2s = 39 99 - 10s = 39 -10s = -60 s = 6, so a = 33 - 24 = 9.
Check: 3(9) + 2(6) = 27 + 12 = 39, and 9 + 4(6) = 9 + 24 = 33. Adult tickets are $9; student tickets are $6.
When Desmos helps versus when algebra is enough
| Situation | Prefer |
|---|---|
| Integer coefficients, SPR, exact fraction | Algebra (substitution or elimination), then Desmos to check |
| Both equations already in y = mx + b | Set the right sides equal, or graph |
| You suspect parallel or duplicate lines | Compare slopes and intercepts; graph to confirm |
| Arithmetic is getting sloppy | Graph both, read the intersection, then substitute back into the originals |
Never skip the substitute-back check. A graph that looks like (3, 5) still has to satisfy both typed equations. Desmos habits return in 14.3 Reference Sheet, Desmos, and What This Test Does Not Include.
What is the solution of the system x + 2y = 10 and x - y = 1?
Which statement describes the system 2x + y = 7 and 4x + 2y = 10?
A cafe sells smoothies for s dollars each and muffins for m dollars each. Two smoothies and four muffins cost $26. One smoothie and one muffin together cost $8. What is the price of one smoothie?