Systems of linear equations

1. Systems of linear equations

SAT Clarity teaching kit
Original supplementary instruction

Teaching notes: Introduce the skill. These materials are independent of College Board.

2. Learning goal

Choose substitution or elimination, interpret intersections, and distinguish one, no and infinitely many solutions.
Before starting: Solve one-variable equations and distribute factors

Teaching notes: Ask learners to restate the goal. Check the prerequisite before proceeding.

3. Diagnostic

Solve x + y = 7 and x - y = 1. What is x?
A. 3
B. 4
C. 6
D. 8

Teaching notes: Allow quiet thinking before sharing. Record reasoning rather than just the letter.

4. Diagnostic reasoning

Add equations: 2x = 8, so x = 4.

Teaching notes: Answer: B. A: 3 is y in the solution. B: Add equations: 2x = 8, so x = 4. C: 6 is neither the x nor the y value. D: 8 is 2x, not x.

5. A simultaneous solution

A solution must satisfy both equations. For x + y = 7 and x - y = 1, (4, 3) satisfies each.

1. x + y = 7
2. x - y = 1
3. Both hold at (4, 3)

Teaching notes: Ask for an explanation in the learner’s own words. Connect this concept to the next worked example.

6. Substitution

When one variable is isolated, replace that variable in the other equation with its equal expression.

1. y = 2x + 1
2. x + (2x + 1) = 10
3. x = 3 and y = 7

Teaching notes: Ask for an explanation in the learner’s own words. Connect this concept to the next worked example.

7. Elimination

Add or subtract equations to cancel a variable. Multiply an entire equation first when coefficients need to match.

1. 2x + 3y = 13
2. Subtract 2x - y = 1
3. 4y = 12

Teaching notes: Ask for an explanation in the learner’s own words. Connect this concept to the next worked example.

8. Graphical interpretation

Two lines with different slopes intersect once. Parallel distinct lines have no intersection. Coincident lines represent infinitely many solutions.

1. Different slopes
2. One intersection
3. One simultaneous solution

Teaching notes: Ask for an explanation in the learner’s own words. Connect this concept to the next worked example.

9. Applications

Define variables with units. A count equation and a cost equation constrain the same quantities.

1. a + s = 10
2. 8a + 5s = 65
3. a = 5 and s = 5

Teaching notes: Ask for an explanation in the learner’s own words. Connect this concept to the next worked example.

10. Worked example 1

Solve y = 2x + 1 and x + y = 10. What is x?
A. 2
B. 3
C. 4
D. 9

Teaching notes: Model reading the prompt and identifying the requested quantity or relationship before solving.

11. Example 1 reasoning

Substitute: x + 2x + 1 = 10. 3x = 9, so x = 3.

Teaching notes: Answer: B. A: 2 gives x + y = 7. B: Substitute: x + 2x + 1 = 10. 3x = 9, so x = 3. C: 4 gives x + y = 13. D: 9 is 3x before division.

12. Worked example 2

For 2x + 3y = 13 and 2x - y = 1, what is y?
A. 2
B. 3
C. 4
D. 12

Teaching notes: Model reading the prompt and identifying the requested quantity or relationship before solving.

13. Example 2 reasoning

Subtract the second equation: 4y = 12, so y = 3.

Teaching notes: Answer: B. A: 2 makes the equation difference 8 instead of 12. B: Subtract the second equation: 4y = 12, so y = 3. C: 4 is the coefficient of y after subtraction. D: 12 is 4y, not y.

14. Worked example 3

An event sells 10 tickets. Adult tickets cost $8, student tickets cost $5, and revenue is $65. How many adult tickets were sold?
A. 3
B. 5
C. 8
D. 10

Teaching notes: Model reading the prompt and identifying the requested quantity or relationship before solving.

15. Example 3 reasoning

a + s = 10 and 8a + 5s = 65. Substitute s = 10 - a: 3a = 15, so a = 5.

Teaching notes: Answer: B. A: 3 adults and 7 students give $59. B: a + s = 10 and 8a + 5s = 65. Substitute s = 10 - a: 3a = 15, so a = 5. C: 8 adults and 2 students give $74. D: 10 adults give $80.

16. Partial multiplication

Multiplying an equation by 2 requires multiplying every term on both sides.

Teaching notes: Ask students to explain how the error changes the result. Use a counterexample if needed.

17. Sign errors

Subtracting an equation changes every sign in that equation.

Teaching notes: Ask students to explain how the error changes the result. Use a counterexample if needed.

18. One-equation checking

A point on one line can fail the other equation. Check both.

Teaching notes: Ask students to explain how the error changes the result. Use a counterexample if needed.

19. Intersection versus intercept

An axis intercept is not necessarily the intersection of the two given lines.

Teaching notes: Ask students to explain how the error changes the result. Use a counterexample if needed.

20. Guided practice 1

How many solutions do y = 2x + 1 and y = 2x - 3 have?
A. None
B. One
C. Two
D. Infinitely many

Teaching notes: Learners solve first. Ask partners to justify a choice and reject an alternative.

21. Guided practice 1 reasoning

Equal slopes and different intercepts describe distinct parallel lines.

Teaching notes: Answer: A. A: Equal slopes and different intercepts describe distinct parallel lines. B: The lines never intersect. C: Two straight lines cannot have exactly two intersections. D: Different intercepts mean the lines are not coincident.

22. Guided practice 2

How many solutions do x + y = 4 and 2x + 2y = 8 have?
A. None
B. One
C. Two
D. Infinitely many

Teaching notes: Learners solve first. Ask partners to justify a choice and reject an alternative.

23. Guided practice 2 reasoning

Dividing the second equation by 2 gives the first equation.

Teaching notes: Answer: D. A: The equations are consistent. B: The second equation repeats the first constraint. C: Every point on the line works, not exactly two. D: Dividing the second equation by 2 gives the first equation.

24. Guided practice 3

For x + y = 9 and 2x - y = 6, what is x + 2y?
A. 5
B. 8
C. 13
D. 18

Teaching notes: Learners solve first. Ask partners to justify a choice and reject an alternative.

25. Guided practice 3 reasoning

Add equations: 3x = 15, x = 5. y = 4. x + 2y = 13.

Teaching notes: Answer: C. A: 5 is x, not the requested expression. B: 8 is 2y, without x. C: Add equations: 3x = 15, x = 5. y = 4. x + 2y = 13. D: 18 doubles the first equation total rather than the requested expression.

26. Independent practice 1

Which ordered pair satisfies x - y = 2 and 3x + y = 10?
A. (3, 1)
B. (1, 3)
C. (2, 0)
D. (4, 2)

Teaching notes: Answer: A. A: 3 - 1 = 2 and 9 + 1 = 10. B: 1 - 3 is -2, not 2. C: The first equation works but 6 + 0 is not 10. D: The first equation works but 12 + 2 is not 10.

27. Independent practice 2

For x/2 + y = 5 and x - y = 1, what is x?
A. 2
B. 3
C. 4
D. 6

Teaching notes: Answer: C. A: 2 gives y = 1 and the first side 2, not 5. B: 3 gives y = 2 and the first side 3.5. C: Substitute y = x - 1: 1.5x - 1 = 5. Thus x = 4. D: 6 is 1.5x, not x.

28. Independent practice 3

For 2x + ky = 6 and 4x + 6y = 12, which k gives infinitely many solutions?
A. 1
B. 2
C. 3
D. 6

Teaching notes: Answer: C. A: Doubling gives y coefficient 2, not 6. B: Doubling gives y coefficient 4, not 6. C: Doubling the first equation gives 4x + 2ky = 12. Set 2k = 6, so k = 3. D: 6 forgets that the first equation must be doubled.

29. Exit ticket

For y = x + 2 and y = -x + 6, what is their intersection?
A. (2, 4)
B. (4, 2)
C. (0, 2)
D. (0, 6)

Teaching notes: Answer: A. A: Equate expressions: x + 2 = -x + 6. x = 2 and y = 4. B: This reverses x and y and fails y = x + 2. C: This is only the first line’s y-intercept. D: This is only the second line’s y-intercept. Ask the learner to name one remaining uncertainty.

30. Review and next practice

Independent answers: 1: A, 2: C, 3: C
Return to the concept behind each missed item.
Use the extension question in the worksheet.
Version 1.0, editorial review 2026-09-17

Teaching notes: Review the separate teacher edition for all distractor explanations. Domain mapping source: https://satsuite.collegeboard.org/higher-ed-professionals/sat-validity/content-domains . Difficulty estimates are instructional, not empirically calibrated.