Question

directions: write an equation passing through the point and parallel to the given line.

1. (4, 7); ( y = 3x + 6 )
2. (-2, 3); ( y = x + 4 )
3. (-4, -5); ( y = \frac{1}{2}x - 6 )
4. (-8, 2); ( 5x - 4y = 4 )
5. (-10, 1); ( 2x + 5y = 15 )
6. (-5, -1); ( 2y = 2x - 4 )

Explanation:

Problem 1: \((4, 7)\); \(y = 3x + 6\)

Step 1: Determine the slope of the parallel line

Parallel lines have the same slope. The given line \(y = 3x + 6\) is in slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept). So, the slope \(m\) of the given line is \(3\). Therefore, the slope of the line parallel to it is also \(m = 3\).

Step 2: Use the point - slope form to find the equation of the line

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. We have the point \((4,7)\) (so \(x_1 = 4\) and \(y_1=7\)) and \(m = 3\).
Substitute these values into the point - slope form:
\(y-7 = 3(x - 4)\)

Step 3: Simplify the equation to slope - intercept form

Expand the right - hand side: \(y-7=3x-12\)
Add \(7\) to both sides of the equation: \(y=3x-12 + 7\)
Simplify the right - hand side: \(y=3x-5\)

Step 1: Determine the slope of the parallel line

The given line \(y=x + 4\) is in slope - intercept form \(y = mx + b\). The slope \(m\) of the given line is \(1\). So, the slope of the line parallel to it is also \(m = 1\).

Step 2: Use the point - slope form to find the equation of the line

Using the point - slope form \(y - y_1=m(x - x_1)\) with the point \((-2,3)\) (where \(x_1=-2\) and \(y_1 = 3\)) and \(m = 1\):
\(y - 3=1\times(x-(-2))\)
\(y - 3=x + 2\)

Step 3: Simplify the equation to slope - intercept form

Add \(3\) to both sides: \(y=x+2 + 3\)
\(y=x + 5\)

Step 1: Determine the slope of the parallel line

The given line \(y=\frac{1}{2}x-6\) is in slope - intercept form. The slope \(m\) of the given line is \(\frac{1}{2}\). So, the slope of the line parallel to it is also \(m=\frac{1}{2}\).

Step 2: Use the point - slope form to find the equation of the line

Using the point - slope form \(y - y_1=m(x - x_1)\) with the point \((-4,-5)\) (where \(x_1=-4\) and \(y_1=-5\)) and \(m = \frac{1}{2}\):
\(y-(-5)=\frac{1}{2}(x-(-4))\)
\(y + 5=\frac{1}{2}(x + 4)\)

Step 3: Simplify the equation to slope - intercept form

Expand the right - hand side: \(y+5=\frac{1}{2}x+2\)
Subtract \(5\) from both sides: \(y=\frac{1}{2}x+2 - 5\)
\(y=\frac{1}{2}x-3\)

Answer:

\(y = 3x-5\)

Problem 2: \((-2, 3)\); \(y=x + 4\)