Linear Functions and Their Graphs · Sub-skill drill
Writing Equations from Points and Slopes
Given a slope and a point, the fastest equation form is point-slope: y − y_1 = m(x − x_1). Given two points, compute the slope first, then plug into point-slope. Avoid jumping straight to slope-intercept form unless the y-intercept is one of the given points; the algebra to extract b from a non-intercept point introduces extra arithmetic and an extra place to make a sign error. SAT wrong-answer choices for these problems are usually the values that appear if you misuse the slope formula by reversing y-values and x-values.
How this sub-skill is tested on the SAT
Given a slope and a point, the fastest equation form is point-slope: y − y_1 = m(x − x_1). Given two points, compute the slope first, then plug into point-slope. Avoid jumping straight to slope-intercept form unless the y-intercept is one of the given points; the algebra to extract b from a non-intercept point introduces extra arithmetic and an extra place to make a sign error. SAT wrong-answer choices for these problems are usually the values that appear if you misuse the slope formula by reversing y-values and x-values.
This sub-skill sits inside the broader Linear Functions and Their Graphs topic, which is part of the College Board's Heart of Algebra content domain. Heart of Algebra accounts for roughly a third of every SAT Math section. The College Board frames it as the ability to analyze, fluently solve, and create linear equations and inequalities — and to interpret what their solutions mean in context. If you walk into test day weak here, no amount of advanced math fluency will compensate, because Heart of Algebra questions appear in both the calculator and no-calculator modules and are usually front-loaded so they set the tempo for the entire section. Mastery looks like solving a two-step linear equation in under twenty seconds, recognizing parall
Practice questions in this drill set
Below are 6 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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A phone plan charges a monthly base of $14 plus $13 per gigabytes used. Which equation gives the monthly bill y for x gigabytes used?
- A y = 13x + 14
- B y = 27x
- C y = 13x - 14
- D y = 14x + 13
Worked solution
Answer: A — y = 13x + 14
The fixed monthly base of $14 is the y-intercept (the cost when x = 0). The per-gigabytes used rate $13 is the slope (the change per unit of x). Combining them gives y = 13x + 14. Sanity check: at x = 3, y = 13(3) + 14 = 53.
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A plumber charges a service call fee of $7 plus $15 per hours. Which equation gives the total bill y for x hours?
- A y = 7x + 15
- B y = 15x - 7
- C y = 22x
- D y = 15x + 7
Worked solution
Answer: D — y = 15x + 7
The fixed service call fee of $7 is the y-intercept (the cost when x = 0). The per-hours rate $15 is the slope (the change per unit of x). Combining them gives y = 15x + 7. Sanity check: at x = 12, y = 15(12) + 7 = 187.
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A gym charges a monthly fee of $49 plus $9 per visits. Which equation gives the total cost y for x visits?
- A y = 9x - 49
- B y = 58x
- C y = 49x + 9
- D y = 9x + 49
Worked solution
Answer: D — y = 9x + 49
The fixed monthly fee of $49 is the y-intercept (the cost when x = 0). The per-visits rate $9 is the slope (the change per unit of x). Combining them gives y = 9x + 49. Sanity check: at x = 12, y = 9(12) + 49 = 157.
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A printer charges a setup charge of $37 plus $10 per pages. Which equation gives the order cost y for x pages?
- A y = 10x + 37
- B y = 10x - 37
- C y = 37x + 10
- D y = 47x
Worked solution
Answer: A — y = 10x + 37
The fixed setup charge of $37 is the y-intercept (the cost when x = 0). The per-pages rate $10 is the slope (the change per unit of x). Combining them gives y = 10x + 37. Sanity check: at x = 4, y = 10(4) + 37 = 77.
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A rideshare charges a pickup fee of $44 plus $6 per miles. Which equation gives the fare y for x miles?
- A y = 6x - 44
- B y = 6x + 44
- C y = 44x + 6
- D y = 50x
Worked solution
Answer: B — y = 6x + 44
The fixed pickup fee of $44 is the y-intercept (the cost when x = 0). The per-miles rate $6 is the slope (the change per unit of x). Combining them gives y = 6x + 44. Sanity check: at x = 19, y = 6(19) + 44 = 158.
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A phone plan charges a monthly base of $21 plus $13 per gigabytes used. Which equation gives the monthly bill y for x gigabytes used?
- A y = 21x + 13
- B y = 13x - 21
- C y = 34x
- D y = 13x + 21
Worked solution
Answer: D — y = 13x + 21
The fixed monthly base of $21 is the y-intercept (the cost when x = 0). The per-gigabytes used rate $13 is the slope (the change per unit of x). Combining them gives y = 13x + 21. Sanity check: at x = 4, y = 13(4) + 21 = 73.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.