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1. write the equation of a line with a slope of 3 and a y - intercept of - 2 in slope - intercept form
2. write the equation of a line with a slope of - 7 and a y - intercept of 12 in slope - intercept form
3. write an equation of a line with a slope of - 2 that goes through (-3,1) in slope - intercept form.
4. write an equation of a line with a slope of 2 that goes through (1,-5) in slope - intercept form.
5. write an equation of a line that goes through (1,0) and (0,2) in slope - intercept form.
6. write an equation of a line that goes through (-1,1) and (-3,-3) in slope - intercept form.

Explanation:

Step1: Recall the slope - intercept form

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.

Step2: Solve problem 1

Given \(m = 3\) and \(b=-2\), substitute into \(y=mx + b\). So \(y = 3x-2\).

Step3: Solve problem 2

Given \(m=-7\) and \(b = 12\), substitute into \(y=mx + b\). So \(y=-7x + 12\).

Step4: Solve problem 3

Use the point - slope form \(y - y_1=m(x - x_1)\). Here \(m=-2\), \(x_1=-3\), \(y_1 = 1\).
\(y - 1=-2(x+3)\)
\(y-1=-2x-6\)
\(y=-2x-5\)

Step5: Solve problem 4

Use the point - slope form \(y - y_1=m(x - x_1)\). Here \(m = 2\), \(x_1=1\), \(y_1=-5\).
\(y + 5=2(x - 1)\)
\(y+5=2x-2\)
\(y = 2x-7\)

Step6: Solve problem 5

First, find the slope \(m=\frac{y_2-y_1}{x_2-x_1}=\frac{2 - 0}{0 - 1}=-2\). The \(y\) - intercept \(b = 2\) (from the point \((0,2)\)).
Substitute \(m=-2\) and \(b = 2\) into \(y=mx + b\). So \(y=-2x + 2\).

Step7: Solve problem 6

First, find the slope \(m=\frac{y_2-y_1}{x_2-x_1}=\frac{-3 - 1}{-3+1}=\frac{-4}{-2}=2\).
Use the point - slope form with the point \((-1,1)\) and \(m = 2\):
\(y - 1=2(x + 1)\)
\(y-1=2x+2\)
\(y = 2x+3\)

Answer:

1. \(y = 3x - 2\)
2. \(y=-7x + 12\)
3. \(y=-2x - 5\)
4. \(y = 2x-7\)
5. \(y=-2x + 2\)
6. \(y = 2x+3\)