Point-slope Form Equations Can Be Converted Into Slope-intercept Form. Determine Whether The Following Equations Represent The Same Line. Prove And Explain Your Answer. Show Your Work For Your Teacher To Review.${ Y + 1 = 5(x + 2) }$ $[

Mar 9, 2025 by ADMIN 238 views

**Converting Point-Slope Form Equations to Slope-Intercept Form**

Introduction

In mathematics, the slope-intercept form of a linear equation is a fundamental concept used to represent lines on a coordinate plane. The slope-intercept form is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. On the other hand, the point-slope form is another way to represent a line, given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. In this article, we will explore how to convert point-slope form equations into slope-intercept form and determine whether two given equations represent the same line.

Converting Point-Slope Form to Slope-Intercept Form

To convert a point-slope form equation into slope-intercept form, we need to isolate the variable y. The general steps to follow are:

Distribute the slope (m) to the terms inside the parentheses.
Add or subtract the constant term (y1) to both sides of the equation.
Simplify the equation to obtain the slope-intercept form.

Let's apply these steps to the given equation:

${ y + 1 = 5(x + 2) }$

Step 1: Distribute the slope (m)

Distribute the slope (5) to the terms inside the parentheses:

${ y + 1 = 5x + 10 }$

Step 2: Add or subtract the constant term (y1)

Subtract 1 from both sides of the equation to isolate the variable y:

${ y = 5x + 9 }$

Step 3: Simplify the equation

The equation is now in slope-intercept form, where the slope (m) is 5 and the y-intercept (b) is 9.

Determining Whether Two Equations Represent the Same Line

To determine whether two equations represent the same line, we need to compare their slopes and y-intercepts. If the slopes and y-intercepts are the same, then the two equations represent the same line.

Let's consider the two equations:

${ y + 1 = 5(x + 2) }$ ${ y = 2x + 3 }$

Step 1: Convert the second equation to slope-intercept form

To compare the two equations, we need to convert the second equation to slope-intercept form:

${ y = 2x + 3 }$

This equation is already in slope-intercept form, where the slope (m) is 2 and the y-intercept (b) is 3.

Step 2: Compare the slopes and y-intercepts

Compare the slopes and y-intercepts of the two equations:

Equation 1: m = 5, b = 9 Equation 2: m = 2, b = 3

Since the slopes (m) are not the same (5 ≠ 2), the two equations do not represent the same line.

Conclusion

In conclusion, we have learned how to convert point-slope form equations into slope-intercept form and determine whether two equations represent the same line. By following the steps outlined in this article, we can easily convert point-slope form equations into slope-intercept form and compare their slopes and y-intercepts to determine whether they represent the same line.

Example Problems

Convert the following point-slope form equation into slope-intercept form:

${ y - 2 = 3(x + 1) }$

Determine whether the following two equations represent the same line:

${ y + 2 = 4(x - 1) }$ ${ y = 2x + 1 }$

Answer Key

y = 3x + 5
No, the two equations do not represent the same line.

Tips and Tricks

When converting point-slope form equations to slope-intercept form, make sure to distribute the slope (m) to the terms inside the parentheses.
When comparing the slopes and y-intercepts of two equations, make sure to check if the slopes (m) and y-intercepts (b) are the same.
If the slopes (m) are not the same, the two equations do not represent the same line.
Point-Slope Form Equations: Q&A =====================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about point-slope form equations.

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Q: How do I convert a point-slope form equation into slope-intercept form?

A: To convert a point-slope form equation into slope-intercept form, follow these steps:

Distribute the slope (m) to the terms inside the parentheses.
Add or subtract the constant term (y1) to both sides of the equation.
Simplify the equation to obtain the slope-intercept form.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is given by the equation y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I determine whether two equations represent the same line?

A: To determine whether two equations represent the same line, compare their slopes and y-intercepts. If the slopes and y-intercepts are the same, then the two equations represent the same line.

Q: What is the difference between the point-slope form and the slope-intercept form?

A: The point-slope form is given by the equation y - y1 = m(x - x1), while the slope-intercept form is given by the equation y = mx + b. The point-slope form is used to find the equation of a line when we know a point on the line and the slope, while the slope-intercept form is used to find the equation of a line when we know the slope and the y-intercept.

Q: Can I use the point-slope form to find the equation of a line when I know the slope and the y-intercept?

A: No, the point-slope form is not used to find the equation of a line when we know the slope and the y-intercept. In that case, we use the slope-intercept form.

Q: Can I use the slope-intercept form to find the equation of a line when I know a point on the line and the slope?

A: Yes, we can use the slope-intercept form to find the equation of a line when we know a point on the line and the slope. We can substitute the point into the equation and solve for the y-intercept.

Q: What is the significance of the slope in a linear equation?

A: The slope in a linear equation represents the rate of change of the line. It tells us how much the line rises or falls for a given change in the x-coordinate.

Q: What is the significance of the y-intercept in a linear equation?

A: The y-intercept in a linear equation represents the point where the line intersects the y-axis. It tells us the value of y when x is equal to 0.

Q: Can I use the point-slope form to find the equation of a line when I know two points on the line?

A: Yes, we can use the point-slope form to find the equation of a line when we know two points on the line. We can use the two points to find the slope and then substitute one of the points into the equation to find the y-intercept.

Q: Can I use the slope-intercept form to find the equation of a line when I know two points on the line?

A: Yes, we can use the slope-intercept form to find the equation of a line when we know two points on the line. We can use the two points to find the slope and then substitute one of the points into the equation to find the y-intercept.