Question

graph the line that passes through the points (2, -6) and (-4, -3) and determine the equation of the line.

Explanation:

Step1: Calculate the slope (m)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1, y_1)=(2, - 6)\) and \((x_2, y_2)=(-4, - 3)\). Then \(m=\frac{-3-(-6)}{-4 - 2}=\frac{-3 + 6}{-6}=\frac{3}{-6}=-\frac{1}{2}\).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\). We can use the point \((2,-6)\) and \(m =-\frac{1}{2}\). Substitute into the formula: \(y-(-6)=-\frac{1}{2}(x - 2)\), which simplifies to \(y + 6=-\frac{1}{2}x + 1\). Then, subtract 6 from both sides: \(y=-\frac{1}{2}x+1 - 6\), so \(y =-\frac{1}{2}x-5\). We can also verify with the other point \((-4,-3)\). Substitute \(x=-4\) into the equation: \(y=-\frac{1}{2}(-4)-5 = 2-5=-3\), which matches the \(y\) - coordinate of the point \((-4,-3)\).

Answer:

The equation of the line is \(y =-\frac{1}{2}x-5\)