Question
Write an equation for line L in point-slope form and slope-intercept form.
L is parallel to y equals 4 x.
-6
-4
-2
2
4
6
-6
-4
-2
2
4
6
x
y
y equals 4 xleft parenthesis negative 3 comma negative 2 right parenthesisUpper L
A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. From left to right, a line rises in quadrant 3 and passes through the origin into quadrant 1 and is labeled y = 4 x. A second line labeled L rises from left to right parallel to the first line passing through the plotted point (negative 3, negative 2).
.
.
.
Question content area right
Part 1
Write an equation for line L in point-slope form.
enter your response here
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Part 2
Write an equation for line L in slope-intercept form.
enter your response here
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
L is parallel to y equals 4 x.
-6
-4
-2
2
4
6
-6
-4
-2
2
4
6
x
y
y equals 4 xleft parenthesis negative 3 comma negative 2 right parenthesisUpper L
A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. From left to right, a line rises in quadrant 3 and passes through the origin into quadrant 1 and is labeled y = 4 x. A second line labeled L rises from left to right parallel to the first line passing through the plotted point (negative 3, negative 2).
.
.
.
Question content area right
Part 1
Write an equation for line L in point-slope form.
enter your response here
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Part 2
Write an equation for line L in slope-intercept form.
enter your response here
(Simplify your answer. Use integers or fractions for any numbers in the equation.)
Answers
Answered by
GPT-4o mini
### Part 1: Equation for Line L in Point-Slope Form
The slope of the line \( y = 4x \) is \( 4 \). Since line L is parallel to this line, it shares the same slope. The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. For line L, using the point \((-3, -2)\):
- \( m = 4 \)
- \( x_1 = -3 \)
- \( y_1 = -2 \)
Plugging in these values, the equation becomes:
\[
y - (-2) = 4(x - (-3))
\]
This simplifies to:
\[
y + 2 = 4(x + 3)
\]
Thus, the equation for line L in point-slope form is:
\[
y + 2 = 4(x + 3)
\]
---
### Part 2: Equation for Line L in Slope-Intercept Form
To convert the point-slope form into slope-intercept form (\( y = mx + b \)), we can start from the point-slope equation we derived:
\[
y + 2 = 4(x + 3)
\]
Distributing \( 4 \) on the right side gives:
\[
y + 2 = 4x + 12
\]
Next, we subtract \( 2 \) from both sides:
\[
y = 4x + 12 - 2
\]
This simplifies to:
\[
y = 4x + 10
\]
Thus, the equation for line L in slope-intercept form is:
\[
y = 4x + 10
\]
---
### Summary
- Point-Slope Form: \( y + 2 = 4(x + 3) \)
- Slope-Intercept Form: \( y = 4x + 10 \)
The slope of the line \( y = 4x \) is \( 4 \). Since line L is parallel to this line, it shares the same slope. The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. For line L, using the point \((-3, -2)\):
- \( m = 4 \)
- \( x_1 = -3 \)
- \( y_1 = -2 \)
Plugging in these values, the equation becomes:
\[
y - (-2) = 4(x - (-3))
\]
This simplifies to:
\[
y + 2 = 4(x + 3)
\]
Thus, the equation for line L in point-slope form is:
\[
y + 2 = 4(x + 3)
\]
---
### Part 2: Equation for Line L in Slope-Intercept Form
To convert the point-slope form into slope-intercept form (\( y = mx + b \)), we can start from the point-slope equation we derived:
\[
y + 2 = 4(x + 3)
\]
Distributing \( 4 \) on the right side gives:
\[
y + 2 = 4x + 12
\]
Next, we subtract \( 2 \) from both sides:
\[
y = 4x + 12 - 2
\]
This simplifies to:
\[
y = 4x + 10
\]
Thus, the equation for line L in slope-intercept form is:
\[
y = 4x + 10
\]
---
### Summary
- Point-Slope Form: \( y + 2 = 4(x + 3) \)
- Slope-Intercept Form: \( y = 4x + 10 \)