To find the linear equation of a line that passes through the points \((-1, 7)\) and \( (2, 4) \) using the point-slope form method, follow these steps:
1. Calculate the slope \( m \) of the line:
The slope \( m \) is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points \((x_1, y_1) = (-1, 7)\) and \((x_2, y_2) = (2, 4)\):
[tex]\[
m = \frac{4 - 7}{2 - (-1)} = \frac{-3}{3} = -1
\][/tex]
2. Use the point-slope form of the equation of a line:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using the slope \( m = -1 \) and one of the given points \((-1, 7)\):
[tex]\[
y - 7 = -1(x - (-1)) \quad \text{or} \quad y - 7 = -1(x + 1)
\][/tex]
Or using the other point \((2, 4)\):
[tex]\[
y - 4 = -1(x - 2)
\][/tex]
3. Convert the equation to slope-intercept form \( y = mx + b \) (optional):
Expanding \( y - 7 = -1(x + 1) \):
[tex]\[
y - 7 = -x - 1 \quad \Rightarrow \quad y = -x - 1 + 7 \quad \Rightarrow \quad y = -x + 6
\][/tex]
Or expanding \( y - 4 = -1(x - 2) \):
[tex]\[
y - 4 = -x + 2 \quad \Rightarrow \quad y = -x + 2 + 4 \quad \Rightarrow \quad y = -x + 6
\][/tex]
Let's examine which statements provided are correct steps in this process:
1. \( y = x + 6 \) — Incorrect: The correct equation in slope-intercept form is \( y = -x + 6 \).
2. \( 7 = -1(-1) + b \) — Correct: This is part of solving for the y-intercept \( b \) using the point \((-1, 7)\).
3. \( y - 4 = -1(x - 2) \) — Correct: This is the point-slope form using the point \((2, 4)\).
4. \( y - 7 = -1(x - (-1)) \) — Correct: This is the point-slope form using the point \((-1, 7)\).
5. \( y - 2 = x - 4 \) — Incorrect: This is not the correct form as it suggests a different slope.
6. \( y = -x + 6 \) — Correct: This is the correct slope-intercept form of the equation.
So, the correct statements are:
- \( 7 = -1(-1) + b \)
- \( y - 4 = -1(x - 2) \)
- \( y - 7 = -1(x - (-1)) \)
- [tex]\( y = -x + 6 \)[/tex]
