➕ Algebra 1 · Linear Functions

Point-Slope Form practice questions

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Answer key with explanations

1. Which choice gives the slope and a point on the line y + 3 = −2(x − 7)?

   Answer: m = −2 and the point (7, −3). The slope is the number in front of the parenthesis, −2. The y + 3 is y − (−3), so y₁ = −3. The x − 7 gives x₁ = 7. The point is (7, −3). Check: at x = 7 the right side is −2(0) = 0, so y = −3. The point (7, 3) forgets to flip the sign of the y number, (−7, −3) also flips the sign of the x number, which should stay 7, and (−3, 7) swaps x and y.

2. A line has slope 3 and passes through (4, −5). Which equation is in point-slope form for this line?

   Answer: y + 5 = 3(x − 4). Here x₁ = 4 and y₁ = −5. So y − y₁ is y − (−5) = y + 5, and x − x₁ is x − 4. The equation is y + 5 = 3(x − 4). Check the point: −5 + 5 = 0 and 3(4 − 4) = 0. The choice y − 5 = 3(x − 4) forgets that y₁ is negative, y + 5 = 3(x + 4) also changes the sign of x₁, and y − 4 = 3(x + 5) swaps x and y.

3. Which equation is the slope-intercept form of y − 3 = 4(x + 2)?

   Answer: y = 4x + 11. Distribute the 4 to both terms: y − 3 = 4x + 8. Add 3 to both sides: y = 4x + 11. Check with the known point (−2, 3): 4(−2) + 11 = 3. The choice y = 4x + 8 forgets to add 3, y = 4x + 5 subtracts 3 instead, and y = 4x − 5 reads x + 2 as x₁ = 2.

4. Which of these is an equation of the line through (−7, 2) and (2, 8)?

   Answer: y − 2 = (2/3)(x + 7). Find the slope first: (8 − 2) ÷ (2 − (−7)) = 6/9 = 2/3. Use (−7, 2) as the known point: y − 2 = (2/3)(x − (−7)), which is y − 2 = (2/3)(x + 7). Check the other point: 8 − 2 = 6 and (2/3)(2 + 7) = 6. (Using (2, 8) instead gives y − 8 = (2/3)(x − 2), the same line, but that form is not one of the choices.) The choice with 3/2 flips the slope, the one with x − 7 forgets the sign of x₁, and the last one uses the point (2, 8) but flips the sign of its x-coordinate.

5. Which point is also on the line y − 1 = −(3/4)(x + 8)?

   Answer: (−4, −2). The line passes through (−8, 1) and has slope −3/4: down 3 for each 4 to the right. Going right 4 from x = −8 reaches x = −4, and going down 3 from y = 1 reaches y = −2. Check: −2 − 1 = −3 and −(3/4)(−4 + 8) = −3. The point (−4, 4) goes up instead of down. The point (−5, −3) swaps the rise and the run. The point (12, −4) comes from misreading the known point as (8, −1).

6. After 6 weeks, Jordan has 85 dollars saved. He adds the same 12 dollars each week. Let w be the number of weeks since he started saving and S his savings in dollars. Which equation models S?

   Answer: S − 85 = 12(w − 6). The slope is the rate, 12 dollars per week. The known reading is 85 dollars at week 6, the point (6, 85). So S − 85 = 12(w − 6). Simplified, S = 12w + 13, so he started with 13 dollars. The choice with S − 6 swaps the two numbers in the point, the choice with S + 85 flips signs that should not change, and the last choice swaps the slope and the week.

7. Which equation is the slope-intercept form of the line through (5, −3) and (−1, 9)?

   Answer: y = −2x + 7. The slope is (9 − (−3)) ÷ (−1 − 5) = 12 ÷ (−6) = −2. Use (5, −3): y + 3 = −2(x − 5). Distribute: y + 3 = −2x + 10, so y = −2x + 7. Check the other point: −2(−1) + 7 = 9. The choice y = 2x − 13 has the wrong sign for the slope (it fits (5, −3) but not (−1, 9)). The choice y = −2x − 13 reads x − 5 as x + 5. The choice y = −2x − 3 uses the y-value of a point as b.

8. Which statement about the line through (5, 1) and (5, 9) is correct?

   Answer: It is the vertical line x = 5. Its slope is undefined, so it has no point-slope equation.. The rise is 9 − 1 = 8 and the run is 5 − 5 = 0, so the slope 8 ÷ 0 is undefined. Both points have x = 5, so the line is vertical, with equation x = 5, and it cannot be written as y − y₁ = m(x − x₁). The equation y − 1 = 8(x − 5) fails for the point (5, 9): the left side is 8 but the right side is 0. A run of 0 makes the slope undefined, not 0, and the line y = 5 does not pass through (5, 1).

9. Maya writes y − 5 = 3(x + 2) for the line through (−2, 5) with slope 3. Leo checks that (0, 11) is on that line, uses it as his known point, and writes y − 11 = 3(x − 0). Which statement is correct?

   Answer: Both equations describe the same line, because any point on a line can be the known point.. The point (0, 11) is on Maya’s line: 11 − 5 = 6 and 3(0 + 2) = 6. Both lines have slope 3 and share that point, so they are the same line. Simplified, both give y = 3x + 11. Point-slope form works with any point on the line, so it does not have to be the first point given or the point (0, b) on the y-axis, and different-looking numbers do not make the equations different.