Question

write an equation for the line that passes through: (-4,-7) and (-6,9)
y = 8x + 32
y = -8x - 39
y = 8x + 39
y = -8x - 32

Explanation:

Step1: Calculate the slope (m)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substituting \((x_1, y_1)=(-4, -7)\) and \((x_2, y_2)=(-6, 9)\), we get:
\(m=\frac{9 - (-7)}{-6 - (-4)}=\frac{9 + 7}{-6 + 4}=\frac{16}{-2}=-8\)

Step2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\). Let's use the point \((-4, -7)\) and \(m = - 8\).
Substitute into the formula: \(y-(-7)=-8(x - (-4))\)
Simplify: \(y + 7=-8(x + 4)\)
Expand the right - hand side: \(y+7=-8x-32\)
Subtract 7 from both sides: \(y=-8x-32 - 7=-8x - 39\)? Wait, no, wait. Wait, \(y+7=-8(x + 4)\)
\(y+7=-8x-32\)
\(y=-8x-32 - 7=-8x-39\)? Wait, no, let's check with the other point \((-6,9)\).
Let's use the slope - intercept form \(y=mx + b\). We know \(m=-8\), so \(y=-8x + b\).
Substitute \(x=-4,y=-7\) into \(y=-8x + b\):
\(-7=-8\times(-4)+b\)
\(-7 = 32 + b\)
\(b=-7 - 32=-39\)? Wait, no, that's wrong. Wait, let's recalculate the slope.
Wait, \((x_1,y_1)=(-4,-7)\), \((x_2,y_2)=(-6,9)\)
\(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{9-(-7)}{-6-(-4)}=\frac{16}{-2}=-8\). That's correct.
Now, using \(y = mx + b\), plug in \((-4,-7)\):
\(-7=-8\times(-4)+b\)
\(-7 = 32 + b\)
\(b=-7 - 32=-39\). So the equation is \(y=-8x-39\)? But wait, let's check the other point \((-6,9)\) in \(y=-8x-39\):
\(y=-8\times(-6)-39=48 - 39 = 9\). Yes, that works. Wait, but let's check the options. The purple option is \(y=-8x - 39\). Wait, but let's check the calculation again.
Wait, when we use the point \((-6,9)\) in \(y=-8x + b\):
\(9=-8\times(-6)+b\)
\(9 = 48 + b\)
\(b=9 - 48=-39\). So the equation is \(y=-8x-39\), which is the purple option. Wait, but let's check the initial calculation of the slope again.
Slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{9 - (-7)}{-6 - (-4)}=\frac{16}{-2}=-8\). Correct.
Then using point - slope with \((-4,-7)\): \(y - (-7)=-8(x - (-4))\)
\(y + 7=-8(x + 4)\)
\(y+7=-8x-32\)
\(y=-8x-32 - 7=-8x-39\). Correct.
Wait, but let's check the fourth option \(y=-8x - 32\). If we plug in \(x=-4\) into \(y=-8x - 32\), we get \(y=-8\times(-4)-32=32 - 32 = 0
eq-7\). So that's wrong.
The purple option is \(y=-8x - 39\). Let's check \(x=-4\): \(y=-8\times(-4)-39=32 - 39=-7\). Correct. \(x=-6\): \(y=-8\times(-6)-39=48 - 39 = 9\). Correct.

Answer:

B. \(y=-8x - 39\)