Question

which equation represents the line that passes through (-6, 7) and (-3, 6)?

$y = -3x + 25$

$y = -\frac{1}{3}x + 5$

$y = -\frac{1}{3}x + 9$

$y = -3x - 11y$

Explanation:

Step1: Find the slope (m)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Given points \((-6, 7)\) and \((-3, 6)\), so \(x_1=-6,y_1 = 7,x_2=-3,y_2=6\).
\(m=\frac{6 - 7}{-3-(-6)}=\frac{-1}{3}=-\frac{1}{3}\)

Step2: Use point - slope form to find the equation

The point - slope form is \(y - y_1=m(x - x_1)\). Let's use the point \((-6,7)\) and \(m =-\frac{1}{3}\).
\(y - 7=-\frac{1}{3}(x+ 6)\)
Expand the right - hand side: \(y-7=-\frac{1}{3}x-2\)
Add 7 to both sides: \(y=-\frac{1}{3}x - 2+7\), so \(y =-\frac{1}{3}x+5\)

We can also check by plugging in the points into the equations.
For the point \((-6,7)\):

• For \(y =-\frac{1}{3}x + 5\), when \(x=-6\), \(y=-\frac{1}{3}\times(-6)+5=2 + 5=7\), which matches.
• For \(y =-\frac{1}{3}x+9\), when \(x = - 6\), \(y=-\frac{1}{3}\times(-6)+9=2 + 9 = 11

eq7\)

• For \(y=-3x + 25\), when \(x=-6\), \(y=-3\times(-6)+25=18 + 25=43

eq7\)

• The equation \(y=-3x-11y\) can be rewritten as \(12y=-3x\) or \(y =-\frac{1}{4}x\), when \(x=-6\), \(y=\frac{3}{2}

eq7\)

Answer:

\(y =-\frac{1}{3}x + 5\) (the second option: \(y =-\frac{1}{3}x + 5\))