SAT linear functions and systems: review and practice

• Explain what the slope and y-intercept mean in a real situation.
• Write a linear function from a table, two points or a description.
• Find the x- and y-intercepts from slope-intercept or standard form.
• Solve a system of two linear equations by substitution, elimination or graphing.
• Decide when a system has one solution, no solution or infinitely many, including when a constant is unknown.

• Stopping at x when the question asks for x + y. Solve for both, then combine. The value of x alone and y alone are usually sitting in the choices.
• Flipping the slope formula. Slope is change in y over change in x. Through (1, 3) and (4, 12), the slope is 9/3 = 3, not 1/3.
• Getting the perpendicular slope half right. A line perpendicular to one with slope 2/3 has slope −3/2. Flipping without changing the sign gives 3/2.
• Dropping a negative in elimination. When you subtract one equation from another, subtract every term, including the constant on the right side.
• Using a point that fits only one equation. Plug a candidate pair into both equations; satisfying the first isn't enough.
• Misreading the intercept. In C = 12t + 30, the 30 is the cost when t = 0 (a flat fee), not the cost per ticket.

1. Question 1 · Easy

   The function R is defined by R(n) = 24.8 − 0.4n, where R(n) is the length of wire, in meters, remaining after a craftsperson makes n ornaments, and n is an integer with 0 ≤ n ≤ 62. According to the model, how many meters of wire are used to make each ornament?

   • A0.4
   • B2.5
   • C24.8
   • D62
   Show the answer

   A. Increasing n by 1 changes R(n) by −0.4 meter, so each ornament uses 0.4 meter of wire. The value 2.5 is the reciprocal rate in ornaments per meter, 24.8 is the starting length, and 62 is the number of ornaments that exhaust the supply.

2. Question 2 · Easy

   x = 0, f(x) = −7. x = 2, f(x) = 1. x = 5, f(x) = 13. For the linear function f, the table gives three values of x and their corresponding values of f(x). Which equation defines f?

   • Af(x) = x/4 − 7
   • Bf(x) = 8x − 7
   • Cf(x) = −7x + 4
   • Df(x) = 4x − 7
   Show the answer

   D. The zero-input row gives an intercept of −7, and the slope is (1 − (−7))/(2 − 0) = 4, so f(x) = 4x − 7; the last row checks because 4(5) − 7 = 13. Choice A reverses the slope ratio, choice B omits division by the input change, and choice C exchanges the two coefficients.

3. Question 3 · Medium

   The function g is defined by g(x) = (−2/5)x + 7. What is the x-intercept of the graph of y = g(x) + 3 in the xy-plane?

   • A(−25, 0)
   • B(10, 0)
   • C(35/2, 0)
   • D(25, 0)
   Show the answer

   D. The new equation is y = (−2/5)x + 10, so 0 = (−2/5)x + 10 gives x = 25 and the x-intercept (25, 0). Choice A reverses the solution’s sign, choice B applies the shift downward, and choice C uses the original zero 35/2.

4. Question 4 · Medium

   (3/4)x − (1/2)y = 13 x + 2y = −4 The solution to the given system of equations is (x, y). What is the value of x + y?

   • A−8
   • B4
   • C12
   • D20
   Show the answer

   B. Multiplying the first equation by 4 gives 3x − 2y = 52; adding the second gives 4x = 48, so x = 12 and y = −8, and therefore x + y = 4. The values −8 and 12 report individual coordinates, while 20 adds 8 instead of −8.

5. Question 5 · Hard

   The measures of the angles of a triangle are x°, y°, and 38°, where x > y > 38. Knowing the value of which of the following expressions would NOT be sufficient to determine the values of x and y?

   • A38 − 6x + 2y − 8y
   • B38 − 6x + 2y − 8x
   • C38 − 6x − 2y − 8y
   • D38 + 6x + 2y − 8y
   Show the answer

   A. The angle sum gives x + y = 142, so 38 − 6x + 2y − 8y = 38 − 6(x + y) = −814 for every allowed triangle and supplies no additional information. Both (x, y) = (80, 62) and (90, 52) satisfy all the conditions and give this same value. The other expressions simplify, respectively, to 322 − 16x, 12x − 814, and 4x − 1382, each of which determines one value of x and hence of y when its value is known.

6. Question 6 · Hard

   In the xy-plane, line j is defined by 3x − ay = 12, where a is a constant with 2 < a < 3. Line k is perpendicular to line j at the x-intercept of line j. Which of the following points could lie on line k?

   • A(−4, 8)
   • B(−4, −6)
   • C(−2, 4)
   • D(−2, 5)
   Show the answer

   D. Line j has x-intercept (4, 0) and slope 3/a, so line k satisfies y = −(a/3)(x − 4). The point (−2, 5) gives 5 = 2a, hence a = 5/2, which satisfies 2 < a < 3. The points (−2, 4) and (−4, 8) require the excluded values 2 and 3, respectively, while (−4, −6) requires a = −9/4, contradicting the positive range.

Type both equations as given, even in standard form like 3x − 2y = 12, and click the intersection to read (x, y). For a table, add a table in Desmos, enter the x₁ and y₁ values, and type y₁ ~ mx₁ + b; Desmos reports m and b. If the question asks for a combination like x + y, add the coordinates you read off.

Ashore's SAT course starts with a 15-minute check that finds the question types costing you the most points, then gives you lessons and practice for each.

• Linear Equations and Inequalities
• Ratios, Rates, Percentages, and Units
• One- and Two-Variable Data
• Probability and Statistical Inference
• Equivalent Expressions and Quadratics
• Nonlinear Functions and Exponentials
• Lines, Angles, and Triangles
• Right Triangles and Trigonometry
• Circles, Area, and Volume
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