Linear Equations Quiz

The 𝘹-intercept of line r in the 𝘹𝘺-plane is 12, and line 𝘳 passes through the point (6, 4). Which of the following coordinate pairs lies on a line that is perpendicular to line 𝘳 and intersects line 𝘳 at its 𝘺-intercept?

(−𝟐, 𝟓) 𝐒𝐭𝐫𝐚𝐭𝐞𝐠𝐢𝐜 𝐀𝐝𝐯𝐢𝐜𝐞: Determine the equation for line 𝘳, and then use that equation to find the equation of the perpendicular line. Plug the choices into that equation to see which one satisfies the equation. 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: Using the formula for slope, m = ʸ²⁻ʸ¹⁄ₓ₂₋ₓ₁, and the points (12,0) and (6,4), the slope of line 𝘳 is ⁰⁻⁴⁄₁₂₋₆ = ⁻⁴⁄₆ = -⅔. So, for the point (12,0), 0 = -⅔(12) + b = -8 + b. Thus, b, the y-intercept, is 8. The perpendicular line will have a slope equal to the negative reciprocal of -⅔, which is ³⁄₂. Since the perpendicular has the same y-intercept, 8, the equation for that line is y = ³⁄₂x + 8. Plug the coordinates of the choices into the equation. For (A), ³⁄₂(-5) + 8 ≠ 2. For (B), ³⁄₂(-2) + 8 = -3 + 8 = 5, so (B) is correct.

The total cost to repair Miranda's car muffler is the cost of the part plus the cost of labor. The graph above shows the cost of repairing Miranda's muffler, c, after n hours of labor. Which of the following equations describes the graph? https://jasper.kaptest.com/content/media/93/383293.8.sat-companion_satmathquiz002_20mod.png

𝐜 = 𝟏𝟎𝟎𝐧 + 𝟏𝟎𝟎 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: According to the graph, the y-intercept is $200. Thus, the cost of the muffler is $200. The line on the graph starts to increase at n = 1 at a rate of $100 per hour, so the cost of labor after the first hour is $100 per hour. The equation can be written as c = 100(n − 1) + 200. This does not match any of the answer choices, so simplify the equation: c = 100n - 1 + 200 c = 100n - 100 + 200 c = 100n + 100 This matches (A).

🆇│🆈 - ̲k̲│̲0̲ - ̲4̲│̲6̲ - ̲6̲│̲1̲2̲ The table above shows the corresponding value of y for a given value of x. The relationship between x and y is linear. If k is a constant, which of the following equations represents the data in the table?

𝐲 + 𝟑𝐱 = −𝟑𝐤 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: Use the data points (−4, 6) and (−6, 12) from the table and the slope-intercept equation to write an equation for the data. The slope is ¹²⁻⁶⁄₋₆₋₍₋₄₎ = ⁶⁄₋₂ = -3. Since the answer choices contain k, plug the point (−k, 0) into y = −3x + b to solve for the y-intercept: 0 = −3(−k)+ b, b = −3k. Thus, equation is y = −3x − 3k. (B) is correct. Alternatively, you can plug in (−k, 0) into each of the answer choices. (A) y + 3x = −k → 0 + 3(−k) ≠ −k; eliminate. (B) y + 3x = −3k → 0 + 3(−k) = −3k; (B) is correct. For the record, (C) 3y + x = −3k → 3(0) + (−k) ≠ −3k and (D) 3y + 3x = −k → 3(0) + 3(−k) ≠ −k.

|−4x+7| = 3 What is the positive difference of the solutions to the equation above?

𝟏.𝟓 𝐨𝐫 𝟑/𝟐 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: Recall that the term within the absolute value signs can be either positive or negative: |𝘢| = |−𝘢| = 𝘢. Set −4𝘹 + 7 equal to ±3 and solve for 𝘹: -4𝘹 + 7 = 3 - 4𝘹 = -4𝘹 = 1 and -4𝘹 + 7 = -3 - 4𝘹 = -10𝘹 = -10 - 4𝘹 = 52 The solutions are 1 and 52 or 2.5. The positive difference is 1 - 2.5 = 1.5. 𝐆𝐫𝐢𝐝 𝐢𝐧 𝟏.𝟓 𝐨𝐫 𝟑/𝟐.

T = 10c + 17a The cost of attending the train show is described by the equation above. T represents the total cost in dollars of c children and a adults attending the train show. If the total cost was $397 and 21 children attended the show, how many adults attended the show?

𝟏𝟏 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: Substitute $397 for T and 21 for c in the given equation and then solve for a: T = 10c + 17a 397 = 10(21) + 17a 397 - 210 = 17a 187 = 17a 11 = a Eleven adults attended the train show. Grid in 11.

The two graphs above show the rates at which two tanks are being emptied. How much longer, in hours, will it take to completely empty tank B than it will take to empty tank A? https://jasper.kaptest.com/content/media/96/383296.0.sat-companion_satmathquiz005_20.png

𝟐 𝐒𝐭𝐫𝐚𝐭𝐞𝐠𝐢𝐜 𝐀𝐝𝐯𝐢𝐜𝐞: Determine the equations for the two lines, then set y equal to 0 for each and find the difference between the two. 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: The y-intercepts can be readily determined from the graph. Select two points on each line and plug them into the formula for slope, m = ʸ²⁻ʸ¹⁄ₓ₂₋ₓ₁, to get the rate at which the tanks empty in gallons per hour. For tank A, that could be m = ²⁰⁰⁰ ⁻ ⁸⁰⁰⁰⁄₆ ₋ ₀ = ⁶⁰⁰⁰⁄₆ = -1,000. For tank B, m = ²⁰⁰⁰⁻⁵⁰⁰⁰⁄₆ ₋ ₀ = ⁻³⁰⁰⁰⁄₆ = -500. Thus, the equation for tank A is y = 8,000 − 1,000x and the equation for tank B is y = 5,000 − 500x. When tank A is empty, that is 0 = 8,000 − 1,000x, so x = 8. For tank B, 0 = 5,000 − 500x, so x = 10. Therefore, it takes 10 − 8 = 2 hours longer to empty tank B. (B) is correct.

The point (9, 𝘱) satisfies the equation 𝘺 = 𝘮𝘹. If m is a constant, what is the value of 𝘹 when y = 3𝘱?

𝟐𝟕 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: Substitute the point (9, p) into the equation 𝘺 = 𝘮𝘹 to get 𝘱 = 𝘮(9). Now substitute 9𝘮 for 𝘱 into 𝘺 = 3𝘱 to get 3(9𝘮) = 27𝘮. Finally, substituting 27𝘮 for 𝘺 into 𝘺 = 𝘮x gives 27𝘮 = 𝘮x. Solving for 𝘹 gives 𝘹= 27. Grid in 27.

Which of the following is an equation for line m in the xy-plane above? https://jasper.kaptest.com/content/media/94/383294.2.sat-companion_satmathquiz003_20mod.png

𝟒𝐱 = 𝐲 + 𝟑 𝐒𝐭𝐫𝐚𝐭𝐞𝐠𝐢𝐜 𝐀𝐝𝐯𝐢𝐜𝐞: Note that some of the choices are not stated in y = mx + b form. Solve for the line in that form, then find the choice that is equivalent. 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: Select two points on the line and plug them into the formula for slope, m = ʸ²⁻ʸ¹⁄ₓ₂₋ₓ₁. Using (1,1) and (0,−3), m = ⁻³ ⁻ ⁽¹⁾⁄₀ ₋ ₁ = ⁻⁴⁄₋₁ = 4, The y-intercept is at −3, so the equation of line m is y = 4x −3. Rearranging the equation gives you 4x = y + 3. This matches (C).

The graph of line 𝘭 is shown in the 𝘹𝘺-plane above. Line 𝘯 is parallel to line 𝘭and passes through the point (2, −2). What is the 𝘹-intercept of line 𝘯? https://jasper.kaptest.com/content/media/00/383300.2.sat-companion_satmathquiz009_20mod.png

𝟔 𝐒𝐭𝐫𝐚𝐭𝐞𝐠𝐢𝐜 𝐀𝐝𝐯𝐢𝐜𝐞: Determine the slope of line 𝘭; line 𝘯 will have the same slope because it is parallel to line 𝘭. Use this slope and the given point to determine the equation for line 𝘯, then solve for the 𝘹-intercept. 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: The formula for slope is 𝘮 = ʸ²⁻ʸ¹⁄ₓ₂₋ₓ₁. Using the points (0, 1) and (−2, 0) on line l, this is ¹⁻⁰⁄₀₋₍₋₂₎ = ½. So, the equation for line 𝘯 is 𝘺 = ˣ⁄₂ + 𝘣. Plugging in the given point (2, −2), yields -2 = ²⁄₂ + 𝘣, so 𝘣 = −2 − 1 = −3. The equation for line 𝘯 is thus 𝘺 = ˣ⁄₂ − 3. Set y to 0 to find the 𝘹-intercept: 0 = ˣ⁄₂ − 3. Thus ˣ⁄₂ = 3, and 𝘹 = 6.

A pharmacy determined that, of all the prescriptions dispensed for a certain medicine last year, ⅒ were dispensed to people younger than 21 years old, ⅕ went to persons ages 21 to 35, and ³⁄₁₀ of the rest of the prescriptions went to customers older than 50. If 343 prescriptions went to people ages 36 to 50, how many total prescriptions for this medicine were given out by the pharmacy last year?

𝟕𝟎𝟎 𝐆𝐞𝐭𝐭𝐢𝐧𝐠 𝐭𝐨 𝐭𝐡𝐞 𝐀𝐧𝐬𝐰𝐞𝐫: First, determine what portion of the prescriptions were given to 36 to 50 years old or older than 50 people. That would be the whole minus the portions for people younger than 21 and from 21 to 35, or 1 - ⅒ - ⅕ = 1 - ⅒ - ⅖ = ⁷⁄₁₀. The question states that ³⁄₁₀ of these prescriptions went to customers older than 50; this is ³⁄₁₀ × ⁷⁄₁₀ = ²¹⁄₁₀₀. So, using x as the total number of prescriptions, set up the equation x = ⅒x + ²⁄₁₀x + 343 + ²¹⁄₁₀₀x. Multiply all terms by 100 to get 100x = 10x + 20x + 34,300 + 21x. This simplifies to 49x = 34,300. Divide both sides by 49 to get x = 700. (B) is correct.