Linear equations and functions

1. Linear equations and functions

SAT Clarity teaching kit
Original supplementary instruction

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

2. Learning goal

Solve linear equations, interpret slope and intercept, and check the requested quantity.
Before starting: Arithmetic, negative numbers and the distributive property

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

3. Diagnostic

If 3x + 6 = 21, what is x?
A. 3
B. 5
C. 7
D. 9

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

4. Diagnostic reasoning

Subtract 6: 3x = 15. Divide by 3: x = 5.

Teaching notes: Answer: B. A: 3 gives 15, not 21. B: Subtract 6: 3x = 15. Divide by 3: x = 5. C: 7 results from dividing 21 before removing the 6. D: 9 gives 33, not 21.

5. Equivalent equations

Applying the same valid operation to both sides preserves the solution. For 3x + 6 = 21, subtract 6 from both sides before dividing by 3.

1. 3x + 6 = 21
2. Subtract 6 on both sides
3. 3x = 15

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

6. Distribution and signs

2(x - 4) = 2x - 8. Multiply each term inside the parentheses. A negative factor changes both signs.

1. 2(x - 4)
2. 2 times x and -4
3. 2x - 8

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

7. Slope and intercept

In y = mx + b, m is the change in y per unit increase in x. b is the value of y when x = 0.

1. y = 3x + 2
2. x rises by 1
3. y rises by 3

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

8. Applications and units

A fixed charge contributes to the intercept. A charge per unit contributes to the slope. State what each variable measures.

1. $15 fixed fee
2. $30 for each hour
3. C = 30h + 15

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

9. Solution count

After simplifying, 0 = 0 means all real values satisfy the equation. A contradiction such as 0 = 5 means no solution.

1. 0 = 0: all real x
2. 0 = 5: no solution
3. x = 5: one solution

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

10. Worked example 1

If 2(x - 4) = 10, what is x?
A. 1
B. 5
C. 9
D. 14

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

11. Example 1 reasoning

Divide by 2: x - 4 = 5. Add 4: x = 9.

Teaching notes: Answer: C. A: 1 gives -6, not 10. B: 5 gives 2, not 10. C: Divide by 2: x - 4 = 5. Add 4: x = 9. D: 14 adds 4 to 10 without first dividing by 2.

12. Worked example 2

A tutor charges a $15 booking fee and $30 per hour. Which equation gives cost C for h hours?
A. C = 15h + 30
B. C = 30h + 15
C. C = 45h
D. C = 30h - 15

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

13. Example 2 reasoning

30h is the hourly charge. Add the one-time $15 fee.

Teaching notes: Answer: B. A: This reverses the fixed fee and hourly rate. B: 30h is the hourly charge. Add the one-time $15 fee. C: This incorrectly charges the booking fee every hour. D: This subtracts a fee that the customer must pay.

14. Worked example 3

A line passes through (1, 2) and (4, 11). What is its slope?
A. 1/3
B. 3
C. 9
D. 13/5

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

15. Example 3 reasoning

Slope = (11 - 2)/(4 - 1) = 9/3 = 3.

Teaching notes: Answer: B. A: 1/3 reverses change in y divided by change in x. B: Slope = (11 - 2)/(4 - 1) = 9/3 = 3. C: 9 is the change in y, without dividing by change in x. D: 13/5 uses sums rather than differences.

16. Incomplete distribution

3(x - 2) equals 3x - 6. Multiplying x alone changes the equation.

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

17. Requested expression

If the prompt asks for 2x, finish by doubling the solved value of x.

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

18. Slope versus intercept

A starting fee and a rate describe different quantities. Use units to distinguish them.

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

19. Answer verification

Substitute into the original equation. Equal values on both sides provide a direct check.

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

20. Guided practice 1

If 5x - 2 = 3x + 8, what is 2x?
A. 5
B. 6
C. 10
D. 20

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

21. Guided practice 1 reasoning

Subtract 3x and add 2: 2x = 10.

Teaching notes: Answer: C. A: 5 is x, while the prompt asks for 2x. B: 6 does not satisfy the equation for 2x. C: Subtract 3x and add 2: 2x = 10. D: 20 doubles the already requested value 2x = 10.

22. Guided practice 2

Which equation describes a line parallel to y = 4x that passes through (2, 5)?
A. y = 4x - 3
B. y = 4x + 3
C. y = 2x + 5
D. y = x/4 + 5

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

23. Guided practice 2 reasoning

Parallel lines have slope 4. 5 = 8 + b gives b = -3.

Teaching notes: Answer: A. A: Parallel lines have slope 4. 5 = 8 + b gives b = -3. B: At x = 2 this gives 11, not 5. C: Slope 2 is not parallel to slope 4. D: Slope 1/4 is not parallel to slope 4.

24. Guided practice 3

How many real solutions does 2(x + 3) = 2x + 6 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.

25. Guided practice 3 reasoning

Both sides simplify to 2x + 6, so every real x works.

Teaching notes: Answer: D. A: Distribution produces an identity, not a contradiction. B: No single x is distinguished by the identity. C: Every real x works, not just two values. D: Both sides simplify to 2x + 6, so every real x works.

26. Independent practice 1

If (x + 3)/4 = 5, what is x?
A. 2
B. 8
C. 17
D. 23

Teaching notes: Answer: C. A: Subtracting before removing the division changes the equation. B: 8 gives 11/4, not 5. C: Multiply by 4: x + 3 = 20. Subtract 3: x = 17. D: 23 adds 3 when it should be subtracted.

27. Independent practice 2

A tank contains 50 liters and drains at 2 liters per minute. Which expression gives its volume after t minutes, before it empties?
A. 50 + 2t
B. 2t - 50
C. 50 - 2t
D. 25 - t

Teaching notes: Answer: C. A: This models filling rather than draining. B: This starts at a negative volume. C: Starting volume minus drained volume is 50 - 2t. D: This gives time remaining, not volume in liters.

28. Independent practice 3

How many real solutions does 3x + 1 = 3x + 4 have?
A. None
B. One
C. Three
D. Infinitely many

Teaching notes: Answer: A. A: Subtracting 3x gives 1 = 4, a contradiction. B: No x can make 1 equal 4. C: The coefficient 3 does not determine solution count. D: An identity would have equal constants, which these do not.

29. Exit ticket

The line y = -2x + 7 has what y-intercept?
A. -2
B. 2
C. 5
D. 7

Teaching notes: Answer: D. A: -2 is the slope. B: 2 is the slope magnitude, not the intercept. C: 5 is y when x = 1, not x = 0. D: At x = 0, y = 7, so the y-intercept is 7. Ask the learner to name one remaining uncertainty.

30. Review and next practice

Independent answers: 1: C, 2: C, 3: A
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.