Match Each Equation To An Equivalent Equation Written In Slope-intercept Form.1. \[$ X - 3y = -6 \$\] - \[$ Y = \frac{1}{3}x + 2 \$\]2. \[$-x + \frac{1}{2}y = 3 \$\] - \[$ Y = 2x + 6 \$\]3. \[$ 2y - 6 = X

Feb 28, 2025 by ADMIN 213 views

**Slope-Intercept Form: A Comprehensive Guide to Matching Equations**

Introduction

In mathematics, the slope-intercept form is a fundamental concept used to represent linear equations. It is a powerful tool for solving problems and understanding the relationship between variables. In this article, we will delve into the world of slope-intercept form and explore how to match equations to their equivalent forms.

What is Slope-Intercept Form?

The slope-intercept form of a linear equation is written as:

y = mx + b

Where:

m is the slope of the line
x is the independent variable
y is the dependent variable
b is the y-intercept

This form is called "slope-intercept" because it shows the slope of the line (m) and the point where the line intersects the y-axis (b).

Matching Equations to Slope-Intercept Form

Now that we have a basic understanding of slope-intercept form, let's move on to matching equations to their equivalent forms.

Equation 1: x - 3y = -6

To match this equation to its equivalent form, we need to isolate y. We can do this by adding 3y to both sides of the equation and then dividing both sides by 3.

x - 3y = -6

Add 3y to both sides:

x = -6 + 3y

Divide both sides by 3:

y = (-6 + x) / 3

Simplify the equation:

y = (-6 + x) / 3

This is equivalent to:

y = (1/3)x - 2

So, the correct match for equation 1 is:

${$ x - 3y = -6 $}$ - ${$ y = \frac{1}{3}x - 2 $}$

Equation 2: -x + (1/2)y = 3

To match this equation to its equivalent form, we need to isolate y. We can do this by adding x to both sides of the equation and then multiplying both sides by 2.

-x + (1/2)y = 3

Add x to both sides:

(1/2)y = 3 + x

Multiply both sides by 2:

y = 2(3 + x)

Simplify the equation:

y = 6 + 2x

This is equivalent to:

y = 2x + 6

So, the correct match for equation 2 is:

${$ -x + \frac{1}{2}y = 3 $}$ - ${$ y = 2x + 6 $}$

Equation 3: 2y - 6 = x

To match this equation to its equivalent form, we need to isolate y. We can do this by adding 6 to both sides of the equation and then dividing both sides by 2.

2y - 6 = x

Add 6 to both sides:

2y = x + 6

Divide both sides by 2:

y = (x + 6) / 2

Simplify the equation:

y = (x + 6) / 2

This is equivalent to:

y = (1/2)x + 3

So, the correct match for equation 3 is:

${$ 2y - 6 = x $}$ - ${$ y = \frac{1}{2}x + 3 $}$

Conclusion

In this article, we have explored the concept of slope-intercept form and how to match equations to their equivalent forms. We have seen how to isolate y and simplify equations to find their equivalent forms. By following these steps, you can confidently match equations to their equivalent forms and solve problems with ease.

Frequently Asked Questions

Q: What is slope-intercept form?

A: Slope-intercept form is a way of writing linear equations in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I match an equation to its equivalent form?

A: To match an equation to its equivalent form, you need to isolate y and simplify the equation. You can do this by adding or subtracting terms from both sides of the equation and then dividing or multiplying both sides by a constant.

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

Q&A: Frequently Asked Questions

Q: What is slope-intercept form?

A: Slope-intercept form is a way of writing linear equations in the form y = mx + b, where m is the slope and b is the y-intercept. This form is called "slope-intercept" because it shows the slope of the line (m) and the point where the line intersects the y-axis (b).

Q: How do I match an equation to its equivalent form?

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

A: Slope-intercept form is a way of writing linear equations in the form y = mx + b, while standard form is a way of writing linear equations in the form ax + by = c. The main difference between the two forms is that slope-intercept form shows the slope and y-intercept, while standard form shows the coefficients of x and y.

Q: How do I find the slope and y-intercept of a linear equation?

A: To find the slope and y-intercept of a linear equation, you need to rewrite the equation in slope-intercept form (y = mx + b). The slope (m) is the coefficient of x, and the y-intercept (b) is the constant term.

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

A: The slope of a linear equation represents the rate of change of the dependent variable (y) with respect to the independent variable (x). A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing.

Q: How do I graph a linear equation in slope-intercept form?

A: To graph a linear equation in slope-intercept form, you need to plot the y-intercept (b) on the y-axis and then use the slope (m) to find the x-coordinate of the point where the line intersects the x-axis.

Q: Can I use slope-intercept form to solve systems of linear equations?

A: Yes, you can use slope-intercept form to solve systems of linear equations. By rewriting each equation in slope-intercept form, you can easily find the point of intersection between the two lines.

Q: What are some common mistakes to avoid when working with slope-intercept form?

A: Some common mistakes to avoid when working with slope-intercept form include:

Forgetting to isolate y
Not simplifying the equation
Using the wrong slope or y-intercept
Not checking for extraneous solutions

Conclusion

In this article, we have explored the concept of slope-intercept form and answered some frequently asked questions. We have seen how to match equations to their equivalent forms, find the slope and y-intercept of a linear equation, and graph a linear equation in slope-intercept form. By following these steps and avoiding common mistakes, you can confidently work with slope-intercept form and solve problems with ease.

Additional Resources

Practice Problems

Match the following equations to their equivalent forms in slope-intercept form:
- ${$ 2x - 3y = 6 $}$
- ${$ y = \frac{2}{3}x + 2 $}$
- ${$ y = -\frac{2}{3}x - 2 $}$
Find the slope and y-intercept of the following linear equation:
- ${$ y = 2x - 3 $}$
Graph the following linear equation in slope-intercept form:
- ${$ y = x + 2 $}$

Answer Key

${$ 2x - 3y = 6 $}$ - ${$ y = \frac{2}{3}x - 2 $}$
Slope: 2, Y-intercept: -3
Graph: y = x + 2