Question

practice c. write an equation for a line parallel to the given equation that contains the given point.
3x + y = -5; (-3,7)

1. y = -x + 4; (-2,5)

x + 5y = 25; (5, -3)

1. y + 2 = \frac{1}{3}(x + 6); (6,0)

Explanation:

Step1: Analyze the first problem (3x + y = -5; (-3,7))

Parallel lines have the same slope. First, rewrite the given equation in slope - intercept form (\(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept).
Starting with \(3x + y=-5\), solve for \(y\): \(y=-3x - 5\). So the slope \(m=-3\).
Now, use the point - slope form of a line, which is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(-3,7)\) and \(m = - 3\).
Substitute the values into the point - slope form: \(y - 7=-3(x+3)\).
Expand the right - hand side: \(y - 7=-3x-9\).
Add 7 to both sides: \(y=-3x - 9 + 7=-3x-2\).

Step2: Analyze the second problem (y = -x + 4; (-2,5))

The given line is in slope - intercept form \(y=-x + 4\), so the slope \(m=-1\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-2,5)\) and \(m=-1\).
Substitute the values: \(y - 5=-1(x + 2)\).
Expand the right - hand side: \(y - 5=-x-2\).
Add 5 to both sides: \(y=-x-2 + 5=-x + 3\).

Step3: Analyze the third problem (x + 5y = 25; (5,-3))

First, rewrite the given equation in slope - intercept form. Solve \(x + 5y = 25\) for \(y\):
\(5y=-x + 25\), then \(y=-\frac{1}{5}x + 5\). The slope \(m =-\frac{1}{5}\).
Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(5,-3)\) and \(m =-\frac{1}{5}\).
Substitute the values: \(y+3=-\frac{1}{5}(x - 5)\).
Expand the right - hand side: \(y+3=-\frac{1}{5}x + 1\).
Subtract 3 from both sides: \(y=-\frac{1}{5}x+1 - 3=-\frac{1}{5}x-2\).

Step4: Analyze the fourth problem (y + 2=\frac{1}{3}(x + 6); (6,0))

The given line is in point - slope form \(y - y_1=m(x - x_1)\), where the slope \(m=\frac{1}{3}\).
Using the point - slope form with \((x_1,y_1)=(6,0)\) and \(m=\frac{1}{3}\).
Substitute the values: \(y-0=\frac{1}{3}(x - 6)\).
Simplify: \(y=\frac{1}{3}x-2\).

Answer:

1. For \(3x + y=-5\) and \((-3,7)\): \(y = - 3x-2\)
2. For \(y=-x + 4\) and \((-2,5)\): \(y=-x + 3\)
3. For \(x + 5y = 25\) and \((5,-3)\): \(y=-\frac{1}{5}x-2\)
4. For \(y + 2=\frac{1}{3}(x + 6)\) and \((6,0)\): \(y=\frac{1}{3}x-2\)