Question

Analyzing Steps to Write a Linear Equation Given Two Points
Helena wrote the equation using point-slope form for the line that passes through the points (5, 1) and (3, 5). Analyze the steps of her work using the point-slope form to write the equation.

1. m = StartFraction 5 minus 1 Over 3 minus 5 EndFraction = StartFraction 4 Over negative 2 EndFraction = negative 2. 2. y minus 1 = negative 2 (x minus 3). Y minus 1 = negative 2 x + 6. y minus 1 + 1 = negative 2 x + 6 + 1. 3. y = negative 2 x + 7.

In which step did Helena make an error?
In step 1, she substituted the x values for y and the y values for x.
In step 2, she didn’t use an x and y from the same coordinate pair.
In step 3, she solved for the wrong variable.
Helena did not make an error.

Answer 1

Answer: In step 2.

Explanation: Step 1 is correct: m = (5 − 1)/(3 − 5) = 4/(−2) = −2. In step 2 Helena mixed coordinates: she used y − 1 (from the point (5, 1)) but paired it with (x − 3) (from the point (3, 5)). You must keep x and y from the same point. Correct point-slope forms:

- Using (5,1): y − 1 = −2(x − 5) → y = −2x + 11
- Using (3,5): y − 5 = −2(x − 3) → y = −2x + 11

So the correct line is y = −2x + 11.