Question

write the point - slope form, the slope - intercept form and the standard form for the line that passes through each pair of points.

1. (3,4), (-1,-4)

m =

point - slope form:

slope - intercept form:

standard form:

Explanation:

Step1: Calculate the slope (m)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Given points \((3, 4)\) and \((-1, -4)\), let \(x_1 = 3\), \(y_1 = 4\), \(x_2=-1\), \(y_2 = -4\). Then \(m=\frac{-4 - 4}{-1 - 3}=\frac{-8}{-4}=2\).

Step2: Write point - slope form

The point - slope form of a line is \(y - y_1=m(x - x_1)\). Using the point \((3, 4)\) and \(m = 2\), we get \(y - 4=2(x - 3)\). We could also use the point \((-1,-4)\), then \(y+4 = 2(x + 1)\).

Step3: Convert to slope - intercept form (\(y=mx + b\))

Starting with \(y - 4=2(x - 3)\), expand the right - hand side: \(y-4 = 2x-6\). Then add 4 to both sides: \(y=2x-6 + 4=2x-2\).

Step4: Convert to standard form (\(Ax+By = C\), \(A\geq0\), \(A\) and \(B\) not both zero)

Starting with \(y=2x-2\), subtract \(2x\) from both sides: \(-2x + y=-2\). Multiply both sides by - 1 to make the coefficient of \(x\) positive: \(2x-y = 2\).

Answer:

• \(m = 2\)
• Point - slope form: \(y - 4=2(x - 3)\) (or \(y + 4=2(x + 1)\))
• Slope - intercept form: \(y=2x-2\)
• Standard form: \(2x - y=2\)