Algebra · Math
Linear equations in two variables
Lines in the xy-plane: slope, intercepts, standard form, parallel and perpendicular relationships, and writing the equation of a line through given points.
32 questions in the bank · 6 easier · 21 medium · 5 harder · this domain is ≈35% of the section
What to watch for
From Ax + By = C the slope is −A/B; the minus sign is the most commonly dropped step. Parallel lines share a slope, perpendicular slopes multiply to −1.
How the question is worded
These are the actual stems used in the bank — the SAT reuses a small set of wordings, and recognising them saves time on test day.
- The graph of 5x + 2y = 25 in the xy-plane crosses the x-axis at the point (a, 0). What is the value of a?
- The graph of 3x + 7y = 12 in the xy-plane crosses the x-axis at the point (a, 0). What is the value of a?
- The graph of 4x + 7y = 36 in the xy-plane crosses the x-axis at the point (a, 0). What is the value of a?
- The graph of 6x + 7y = 12 in the xy-plane crosses the x-axis at the point (a, 0). What is the value of a?
Worked examples
easyThe graph of 5x + 2y = 25 in the xy-plane crosses the x-axis at the point (a, …
The graph of 5x + 2y = 25 in the xy-plane crosses the x-axis at the point (a, 0). What is the value of a?
Answer: 5
Explanation
On the x-axis, y = 0. Substitute: 5x = 25, so x = 25/5 = 5. The x-intercept is (5, 0).
mediumWhat is the slope of the graph of 7x − 3y = 17 in the xy-plane?
What is the slope of the graph of 7x − 3y = 17 in the xy-plane?
- A. 7/3
- B. −7/3
- C. −3/7
- D. 3/7
Answer: A. 7/3
Explanation
Solve for y: −3y = 17 − 7x, so y = (17 − 7x)/−3. The x-coefficient is −7/−3 = 7/3. From Ax + By = C the slope is always −A/B; the minus sign is what the most tempting wrong answer drops.
hardLine ℓ is defined by y = −5/3x + 8. Line k is perpendicular to line ℓ in the x…
Line ℓ is defined by y = −5/3x + 8. Line k is perpendicular to line ℓ in the xy-plane. What is the slope of line k?
- A. −5/3
- B. 5/3
- C. −3/5
- D. 3/5
Answer: D. 3/5
Explanation
Parallel lines share a slope; perpendicular lines have slopes whose product is −1. Line ℓ has slope −5/3, so line k has slope 3/5 (the negative reciprocal).