A Line Passes Through The Points { (-1, -6)$}$ And { (7, 2) $}$.Find The Slope-intercept Form Of The Equation Of The Line. Then Fill In The Value Of The Slope, { M $}$, And The Value Of The { Y $}$-intercept,
Introduction
In mathematics, the slope-intercept form of a line is a way to express the equation of a line in the form of { y = mx + b $}$, where { m $}$ is the slope of the line and { b $}$ is the { y $}$-intercept. Given two points on a line, we can find the slope-intercept form of the equation of the line using the slope formula and the point-slope form of a line.
The Slope Formula
The slope formula is used to find the slope of a line given two points on the line. The slope formula is given by:
{ m = \frac{y_2 - y_1}{x_2 - x_1} $}$
where { (x_1, y_1) $}$ and { (x_2, y_2) $}$ are the two points on the line.
Finding the Slope
In this problem, we are given two points on the line: { (-1, -6) $}$ and { (7, 2) $}$. We can use the slope formula to find the slope of the line.
{ m = \frac{2 - (-6)}{7 - (-1)} $}$
{ m = \frac{8}{8} $}$
{ m = 1 $}$
The Point-Slope Form
The point-slope form of a line is given by:
{ 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.
Finding the Equation of the Line
We can use the point-slope form to find the equation of the line. We will use the point { (-1, -6) $}$ and the slope { m = 1 $}$.
{ y - (-6) = 1(x - (-1)) $}$
{ y + 6 = x + 1 $}$
Simplifying the Equation
We can simplify the equation by subtracting 6 from both sides.
{ y = x - 5 $}$
The Slope-Intercept Form
The slope-intercept form of the equation of the line is given by:
{ y = mx + b $}$
where { m $}$ is the slope of the line and { b $}$ is the { y $}$-intercept.
Finding the { y $}$-Intercept
We can find the { y $}$-intercept by setting { x = 0 $}$ in the equation.
{ y = 1(0) - 5 $}$
{ y = -5 $}$
Conclusion
In this problem, we found the slope-intercept form of the equation of a line given two points on the line. We used the slope formula to find the slope of the line and the point-slope form to find the equation of the line. We then simplified the equation and found the { y $}$-intercept.
Final Answer
The slope-intercept form of the equation of the line is:
{ y = x - 5 $}$
The value of the slope, { m $}$, is:
{ m = 1 $}$
The value of the { y $}$-intercept, { b $}$, is:
Q&A: Frequently Asked Questions
Q: What is the slope-intercept form of a line? A: The slope-intercept form of a line is a way to express the equation of a line in the form of { y = mx + b $}$, where { m $}$ is the slope of the line and { b $}$ is the { y $}$-intercept.
Q: How do I find the slope of a line given two points? A: You can use the slope formula to find the slope of a line given two points on the line. The slope formula is given by:
{ m = \frac{y_2 - y_1}{x_2 - x_1} $}$
where { (x_1, y_1) $}$ and { (x_2, y_2) $}$ are the two points on the line.
Q: What is the point-slope form of a line? A: The point-slope form of a line is given by:
{ 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.
Q: How do I find the equation of a line given two points and the slope? A: You can use the point-slope form to find the equation of a line given two points and the slope. We will use the point { (-1, -6) $}$ and the slope { m = 1 $}$.
{ y - (-6) = 1(x - (-1)) $}$
{ y + 6 = x + 1 $}$
Q: How do I simplify the equation of a line? A: You can simplify the equation of a line by subtracting 6 from both sides.
{ y = x - 5 $}$
Q: What is the { y $}$-intercept of a line? A: The { y $}$-intercept of a line is the point where the line intersects the { y $}$-axis. You can find the { y $}$-intercept by setting { x = 0 $}$ in the equation.
Q: How do I find the { y $}$-intercept of a line? A: You can find the { y $}$-intercept of a line by setting { x = 0 $}$ in the equation.
{ y = 1(0) - 5 $}$
{ y = -5 $}$
Q: What is the final answer for the slope-intercept form of the equation of the line? A: The final answer for the slope-intercept form of the equation of the line is:
{ y = x - 5 $}$
Q: What is the value of the slope, { m $}$? A: The value of the slope, { m $}$, is:
{ m = 1 $}$
Q: What is the value of the { y $}$-intercept, { b $}$? A: The value of the { y $}$-intercept, { b $}$, is:
{ b = -5 $}$
Conclusion
In this article, we have discussed the slope-intercept form of a line, the slope formula, the point-slope form, and how to find the equation of a line given two points and the slope. We have also answered some frequently asked questions about the slope-intercept form of a line.