Question

6.2 hw: slope-intercept given slope + point date peri
write the slope-intercept form of the equation of the line using the given information. show all
work when appropriate!

1. through: (-3, -1), slope = 2
2. through: (-3, 4), slope = -\frac{1}{3}

3)

1. slope = -\frac{1}{2}, y-intercept = -3

5)
6)

1. through: (-3, -4), slope = \frac{7}{3}
2. through: (-3, 2), slope = -1

Explanation:

Problem 1: through \((-3, -1)\), slope \(= 2\)

Step 1: Recall the point - slope form \(y - y_1=m(x - x_1)\)

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope of the line. Here, \(x_1=-3\), \(y_1 = - 1\) and \(m = 2\). Substitute these values into the point - slope form:
\(y-(-1)=2(x - (-3))\)

Step 2: Simplify the equation to slope - intercept form \(y=mx + b\)

Simplify the left - hand side and the right - hand side of the equation:
\(y + 1=2(x + 3)\)
Expand the right - hand side: \(y+1 = 2x+6\)
Subtract 1 from both sides: \(y=2x+6 - 1\)
\(y=2x + 5\)

Problem 2: through \((-3,4)\), slope \(=-\frac{1}{3}\)

Step 1: Use the point - slope form \(y - y_1=m(x - x_1)\)

Here, \(x_1=-3\), \(y_1 = 4\) and \(m=-\frac{1}{3}\). Substitute these values into the point - slope form:
\(y - 4=-\frac{1}{3}(x-(-3))\)

Step 2: Simplify to slope - intercept form

Simplify the right - hand side: \(y - 4=-\frac{1}{3}(x + 3)\)
Expand the right - hand side: \(y-4=-\frac{1}{3}x-1\)
Add 4 to both sides: \(y=-\frac{1}{3}x-1 + 4\)
\(y=-\frac{1}{3}x+3\)

Problem 4: Slope \(=-\frac{1}{2}\), \(y\) - intercept \(=-3\)

Step 1: Recall the slope - intercept form \(y=mx + b\)

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

Step 2: Substitute \(m=-\frac{1}{2}\) and \(b = - 3\) into the formula

Substitute \(m=-\frac{1}{2}\) and \(b=-3\) into \(y=mx + b\):
\(y=-\frac{1}{2}x-3\)

Problem 7: through \((-3,-4)\), slope \(=\frac{7}{3}\)

Answer:

Step 1: Calculate the slope \(m\)

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \((x_1,y_1)=(0,-1)\) and \((x_2,y_2)=(5,0)\), \(m=\frac{0-(-1)}{5 - 0}=\frac{1}{5}\)

Step 2: Use the slope - intercept form

The \(y\) - intercept \(b=-1\). So the equation is \(y=\frac{1}{5}x-1\)

Problem 6 Answer: \(y=\frac{1}{5}x-1\)