Question

what is the slope - intercept form of the equation of the line containing the points $(-2, 4)$ and $(6, 0)$?\\(\bigcirc\\) $y = -\frac{1}{2}x + 3$\\(\bigcirc\\) $y = -\frac{1}{2}x + 8$\\(\bigcirc\\) $y = -\frac{1}{2}x + 5$\\(\bigcirc\\) $y = \frac{1}{2}x + 5$

Explanation:

Step1: Calculate the slope (m)

The slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For the points \((-2, 4)\) and \((6, 0)\), \(x_1=-2\), \(y_1 = 4\), \(x_2=6\), \(y_2=0\). So \(m=\frac{0 - 4}{6-(-2)}=\frac{-4}{8}=-\frac{1}{2}\).

Step2: Use point - slope form to find the equation

The point - slope form is \(y - y_1=m(x - x_1)\). We can use the point \((6,0)\) (we could also use \((-2,4)\)). Substitute \(m =-\frac{1}{2}\), \(x_1 = 6\), \(y_1=0\) into the formula: \(y-0=-\frac{1}{2}(x - 6)\).

Step3: Simplify the equation to slope - intercept form (\(y=mx + b\))

Simplify \(y=-\frac{1}{2}x+3\)? Wait, no, wait. Wait, if we use the point \((-2,4)\): \(y - 4=-\frac{1}{2}(x+2)\). Expand it: \(y-4=-\frac{1}{2}x - 1\). Then add 4 to both sides: \(y=-\frac{1}{2}x+3\)? Wait, that's not right. Wait, no, let's recalculate. Wait, when we use the point \((6,0)\): \(y=-\frac{1}{2}(x - 6)=-\frac{1}{2}x+3\)? But let's check with the point \((-2,4)\). Plug \(x=-2\) into \(y =-\frac{1}{2}x + 3\), we get \(y=-\frac{1}{2}\times(-2)+3=1 + 3=4\), which matches. Wait, but let's check the other options. Wait, maybe I made a mistake. Wait, the slope is \(-\frac{1}{2}\). Let's check the third option: \(y=-\frac{1}{2}x + 5\). Plug \(x = 6\) into it: \(y=-\frac{1}{2}\times6+5=-3 + 5=2
eq0\). The second option: \(y=-\frac{1}{2}x + 8\), plug \(x = 6\): \(y=-3 + 8=5
eq0\). The fourth option: slope is \(\frac{1}{2}\), which we calculated the slope as \(-\frac{1}{2}\), so it's wrong. Wait, but when we use the point \((-2,4)\) in \(y=-\frac{1}{2}x + 3\), we get \(y=-\frac{1}{2}\times(-2)+3=1 + 3=4\), which is correct. And when we use \(x = 6\), \(y=-\frac{1}{2}\times6+3=-3 + 3=0\), which is also correct. Wait, but the first option is \(y=-\frac{1}{2}x + 3\). But let's check again. Wait, the slope calculation: \(\frac{0 - 4}{6-(-2)}=\frac{-4}{8}=-\frac{1}{2}\), that's correct. Then using point - slope with \((6,0)\): \(y=-\frac{1}{2}(x - 6)=-\frac{1}{2}x+3\). So the correct equation is \(y =-\frac{1}{2}x + 3\).

Answer:

A. \(y =-\frac{1}{2}x + 3\)