➕ Algebra 1 · Linear Functions

Standard Form of a Line practice questions

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Answer key with explanations

1. In the equation 4x − 9y = 36, what are A, B and C?

   Answer: A = 4, B = −9, C = 36. Match the equation to Ax + By = C. The x term is 4x, so A = 4. The y term is −9y, which is (−9)y, so B = −9 (the minus sign stays with the 9). The number on the right is C = 36. Check: at the point (0, −4), −9(−4) = 36, so the y-intercept is C/B = 36 ÷ (−9) = −4.

2. At which point does the line 5x + 2y = 30 cross the x-axis?

   Answer: (6, 0). Points on the x-axis have y = 0. Cover the y term: 5x = 30, so x = 6, and the line crosses at (6, 0). The point (0, 15) is where the line crosses the y-axis (2y = 30). The point (15, 0) divides by 2 instead of 5, and (0, 6) swaps the coordinates.

3. Where does the line 3x − 8y = 24 cross the x-axis and the y-axis, in that order?

   Answer: (8, 0) and (0, −3). Cover the y term: 3x = 24, so x = 8 and the point is (8, 0). Cover the x term: −8y = 24, so y = −3 and the point is (0, −3). The minus sign stays with the 8. The choice with (0, 3) drops that sign, (3, 0) and (0, −8) use the coefficients as intercepts, and the last pair has both signs wrong.

4. Which equation is 7x − 2y = 8 written in slope-intercept form?

   Answer: y = (7/2)x − 4. Subtract 7x from both sides: −2y = −7x + 8. Divide every term by −2: y = (7/2)x − 4. Check with x = 2: y = 7 − 4 = 3, and 7(2) − 2(3) = 14 − 6 = 8. The choice y = −(7/2)x − 4 forgets to change the sign of the x term, y = (7/2)x + 4 forgets to change the sign of the constant, and y = (2/7)x − 4 flips the fraction.

5. Which equation is y = −(2/3)x + 4 written in standard form (integers, A not negative)?

   Answer: 2x + 3y = 12. Multiply every term by 3 to clear the fraction: 3y = −2x + 12. Add 2x to both sides: 2x + 3y = 12. Check with the point (3, 2), which is on the original line since −(2/3)(3) + 4 = 2: 2(3) + 3(2) = 12. The choice 2x − 3y = 12 has the wrong sign on the y term, 3x + 2y = 12 swaps the coefficients, and 2x + 3y = 4 forgets to multiply the 4 by 3.

6. Which equation is y = (5/3)x − 7/2 written in standard form (integers, A not negative)?

   Answer: 10x − 6y = 21. The denominators are 3 and 2, so multiply every term by 6: 6y = 10x − 21. Subtract 10x from both sides: −10x + 6y = −21. Now A = −10 is negative, so multiply every term by −1: 10x − 6y = 21. Check with (3, 3/2), which is on the original line since (5/3)(3) − 7/2 = 3/2: 10(3) − 6(3/2) = 30 − 9 = 21. The choice 5x − 3y = 21 divides A and B by 2 but not C, so it is a different line. The choice 10x − 6y = −21 has the wrong sign on C (it skips multiplying C by −1), and 10x + 6y = 21 has the wrong sign on the y term.

7. A food truck sells tacos for 3 dollars and burritos for 5 dollars. In one rush it takes in exactly 60 dollars. Let t be the number of tacos sold and b the number of burritos sold, so 3t + 5b = 60. What does the point (0, 12) on the graph mean?

   Answer: 12 burritos and no tacos make exactly 60 dollars.. The point (0, 12) has t = 0 and b = 12. Check: 3(0) + 5(12) = 60. So 12 burritos at 5 dollars each, and no tacos, bring in exactly 60 dollars. It is where the graph crosses the vertical (b) axis (the burrito-only case). 12 tacos would bring in only 36 dollars. The price of a burrito is 5 dollars, and the 0 tacos bring in 0 dollars, not 12.

8. What is the slope of the line 6x + 9y = 18?

   Answer: −2/3. Solve for y: 9y = −6x + 18, so y = −(6/9)x + 2 = −(2/3)x + 2. The slope is −2/3. It matches the shortcut −A/B = −6/9 = −2/3. As a check, the intercepts (3, 0) and (0, 2) give slope (2 − 0) ÷ (0 − 3) = −2/3. The choice 2/3 forgets the minus sign, −3/2 flips the fraction, and −6 is just −A.

9. Which equation has a vertical line as its graph?

   Answer: 7x = −21. 7x = −21 has no y term (B = 0), so x = −3. Every point has x = −3 whatever y is, so the line is vertical. The equation 7y = −21 gives y = −3, a horizontal line. The other two have both variables: 7x + 7y = −21 contains both (−3, 0) and (0, −3), and x + y = −21 contains both (−21, 0) and (0, −21), so neither has a single x-value.

10. Which equation is y − 1 = 2(x + 3) written in standard form (integers, A not negative)?

    Answer: 2x − y = −7. Distribute the 2: y − 1 = 2x + 6. Move the terms: −2x + y = 7. Now A = −2 is negative, so multiply every term by −1: 2x − y = −7. Check with the known point (−3, 1): 2(−3) − 1 = −7. The choice 2x − y = 7 has the wrong sign on C, 2x − y = −4 forgets to multiply the 3 by 2, and 2x + y = −7 has the wrong sign on y.

11. Which point is on the line 5x − 3y = −9?

    Answer: (−6, −7). Substitute each point. For (−6, −7): 5(−6) − 3(−7) = −30 + 21 = −9, which matches. For (6, 7): 30 − 21 = 9, the wrong sign. For (−7, −6) (coordinates swapped): −35 + 18 = −17. For (−3, 2): −15 − 6 = −21. Only (−6, −7) makes the equation true.