A line passes through the point (−3, 4) with slope 2. Which equation correctly represents this line in point-slope form?
Ay − 4 = 2(x − 3)
By − 4 = 2(x + 3)
Cy + 4 = 2(x + 3)
Dy − 4 = 2x − 3
The form is y − y₁ = m(x − x₁). With y₁ = 4, m = 2, and x₁ = −3, substituting literally gives y − 4 = 2(x − (−3)), which simplifies to y − 4 = 2(x + 3). Option A makes the classic sign error: it writes x − 3 instead of x + 3, dropping the double negative. The formula demands 'x minus x₁' — subtracting a negative x₁ produces addition.
Question 2 Multiple Choice
A line has slope 3 and passes through the point (2, 7). After converting to slope-intercept form, what is the y-intercept?
A1
B13
C−1
D7
Starting from point-slope form: y − 7 = 3(x − 2). Distribute: y − 7 = 3x − 6. Add 7: y = 3x + 1. The y-intercept is 1. Option D (7) is a common error — students forget to distribute the slope and incorrectly treat the y₁ value as the y-intercept. Option B (13) comes from treating the point as (−2, 7) and adding rather than subtracting.
Question 3 True / False
When writing a line's equation in point-slope form using two given points, it doesn't matter which point you use as (x₁, y₁) — both produce the same line.
TTrue
FFalse
Answer: True
Both points lie on the same line, so plugging either one into y − y₁ = m(x − x₁) produces a different-looking equation that describes the same geometric object. Distributing and simplifying both equations to slope-intercept form yields identical results. Students sometimes believe they must use the 'first' point listed, but the choice is free.
Question 4 True / False
The equation y − 3 = 4(x + 2) means the line passes through the point (2, 3).
TTrue
FFalse
Answer: False
The form is y − y₁ = m(x − x₁), so x₁ is the value being subtracted from x. Since the equation shows (x + 2), that equals (x − (−2)), meaning x₁ = −2, not 2. The line passes through (−2, 3). This is the sign trap: when the x-coordinate is negative, subtracting it produces addition, which students misread as a positive coordinate.
Question 5 Short Answer
Where does point-slope form come from, and what two pieces of information do you need to write a linear equation using it?
Think about your answer, then reveal below.
Model answer: Point-slope form comes directly from the slope definition: m = (y − y₁)/(x − x₁). Multiplying both sides by (x − x₁) gives y − y₁ = m(x − x₁). You need the slope m and any one specific point (x₁, y₁) on the line. If two points are given instead, compute the slope first, then use either point.
Recognizing point-slope form as a rearrangement of the slope definition — not an arbitrary formula — makes it easier to reconstruct and apply correctly. It also reveals why the signs matter: each variable in the formula is subtracted from its corresponding point coordinate, which is why negative coordinates produce addition in the equation.