Question

write the point - slope form, the slope - intercept form and the standard form for the line that passes through the given point with the given slope.

1. j (-6, -3), m = -\frac{1}{2}

point - slope form:

slope - intercept form:

standard form:

Explanation:

Step1: Recall Point - Slope Formula

The point - slope form of a line is given by \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope of the line.
We are given the point \((-6,-3)\) (so \(x_1=-6\) and \(y_1 = - 3\)) and the slope \(m=-\frac{1}{2}\).
Substitute these values into the point - slope formula:
\(y-(-3)=-\frac{1}{2}(x - (-6))\)
Simplify the equation:
\(y + 3=-\frac{1}{2}(x + 6)\)

Step2: Convert to Slope - Intercept Form (\(y=mx + b\))

Start with the point - slope form \(y + 3=-\frac{1}{2}(x + 6)\)
Distribute the \(-\frac{1}{2}\) on the right - hand side:
\(y+3=-\frac{1}{2}x-3\)
Subtract 3 from both sides of the equation to solve for \(y\):
\(y=-\frac{1}{2}x-3 - 3\)
\(y=-\frac{1}{2}x-6\)

Step3: Convert to Standard Form (\(Ax+By = C\), where \(A\), \(B\), and \(C\) are integers and \(A\geq0\))

Start with the slope - intercept form \(y=-\frac{1}{2}x-6\)
Add \(\frac{1}{2}x\) to both sides of the equation:
\(\frac{1}{2}x+y=-6\)
Multiply every term in the equation by 2 to eliminate the fraction:
\(x + 2y=-12\)

Answer:

• Point - slope form: \(y + 3=-\frac{1}{2}(x + 6)\)
• Slope - intercept form: \(y=-\frac{1}{2}x-6\)
• Standard form: \(x + 2y=-12\)