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SAT · Math · Algebra
Linear equations in two variables is part of Algebra in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include two-variable constraint model, intercepts from standard form, write an equation from a context and point on a line.
Common traps: swapping coefficient and variable; interpreting the intercept as a total; treating both items as the same price.
Question 1
4x − 6y = 9 A ball thrown from a stone jetty is tracked with a line of sight given by the equation above in the xy-plane. What is the slope of this line?
−3/2
−2/3
2/3
4
Answer: C. 2/3
Solving for y: −6y = −4x + 9, so y = (2/3)x − 3/2, and the slope is 2/3 (C). A is the negative reciprocal, the slope of a perpendicular line; B misses that dividing by −6 makes the slope positive; D takes the coefficient of x as the slope without dividing by the y-coefficient.
Question 2
In the xy-plane, a line with slope 3 passes through the point (-6, 9). What is the y-intercept of the line?
-27
-9
27
33
Answer: C. 27
Use y = mx + b: 9 = (3)(-6) + b, so 9 = -18 + b and b = 27.
Question 3
In the xy-plane, a line has a y-intercept of 4 and passes through the point (6, 1). Which of the following is an equation of the line?
y = -2x + 4
y = (1/2)x + 4
y = -(1/2)x + 1
y = -(1/2)x + 4
Answer: D. y = -(1/2)x + 4
The y-intercept gives the point (0, 4), so the slope is (1 - 4)/(6 - 0) = -3/6 = -1/2 and the equation is y = -(1/2)x + 4. Check: at x = 6, y = -3 + 4 = 1. A inverts the slope ratio; B drops the negative sign; C uses the y-coordinate of the given point as the intercept.
Question 4
A fence builder spends exactly $1,620 on posts and panels. Each post costs $18 and each panel costs $45, so the purchase satisfies 18p + 45f = 1,620, where p is the number of posts and f is the number of panels. If the builder buys no posts, how many panels does the builder buy?
18
36
45
90
Answer: B. 36
With p = 0, 45f = 1,620, so f = 36. A is the price of a post, the coefficient of p; C is the price of a panel, the coefficient of f; D is the number of posts when no panels are bought.
Question 5
A coffee roastery blends two beans, and the cost C, in dollars per kilogram, of a blend is a linear function of p, the percent of the blend that is Bean A. A blend with 20% Bean A costs 18.40 dollars per kilogram, and a blend with 60% Bean A costs 16.80 dollars per kilogram. What percent of Bean A gives a blend that costs 17.20 dollars per kilogram?
Enter your own answer.
Answer: 50
The slope is (16.80 - 18.40)/(60 - 20) = -0.04 dollars per percentage point. Solve 17.20 = 18.40 - 0.04(p - 20): p - 20 = 30, so p = 50. Wrong routes: 30 is the change from 20%, without adding back the starting 20; -10 uses a slope of +0.04, a sign error.
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