A Line Has A Slope Of $-\frac{2}{3}$ And Passes Through The Point $(9,0$\]. Write Its Equation In Slope-intercept Form.Write Your Answer Using Integers, Proper Fractions, And Improper Fractions In Simplest Form.

Feb 25, 2025 by ADMIN 212 views

**A Line with a Given Slope and Point: Finding the Equation in Slope-Intercept Form**

Introduction

In mathematics, the slope-intercept form of a linear equation is a powerful tool for representing lines on a coordinate plane. It is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will explore how to find the equation of a line with a given slope and a point it passes through, using the slope-intercept form.

The Slope-Intercept Form

The slope-intercept form of a linear equation is given by:

y = mx + b

where:

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

Given Information

We are given that the line has a slope of $-\frac{2}{3}$ and passes through the point $(9,0)$ . This means that the slope-intercept form of the equation will be of the form:

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

Finding the Y-Intercept

To find the y-intercept, we can substitute the given point $(9,0)$ into the equation. This will give us:

0 = $-\frac{2}{3}$ (9) + b

Simplifying the equation, we get:

0 = -6 + b

Adding 6 to both sides, we get:

b = 6

The Equation of the Line

Now that we have found the y-intercept, we can write the equation of the line in slope-intercept form:

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

Interpretation

The equation y = $-\frac{2}{3}$ x + 6 represents a line with a slope of $-\frac{2}{3}$ and a y-intercept of 6. This means that for every 3 units we move to the right, the line will drop 2 units. The y-intercept of 6 means that the line intersects the y-axis at the point (0,6).

Conclusion

In this article, we have seen how to find the equation of a line with a given slope and a point it passes through, using the slope-intercept form. We have used the given slope and point to find the y-intercept, and then written the equation of the line in slope-intercept form. This is a powerful tool for representing lines on a coordinate plane, and is an essential concept in mathematics.

Example Problems

Problem 1

Find the equation of a line with a slope of $\frac{3}{4}$ and a point it passes through (8,12).

Solution

Using the slope-intercept form, we can write the equation as:

y = $\frac{3}{4}$ x + b

Substituting the given point (8,12), we get:

12 = $\frac{3}{4}$ (8) + b

Simplifying the equation, we get:

12 = 6 + b

Subtracting 6 from both sides, we get:

b = 6

The equation of the line is:

y = $\frac{3}{4}$ x + 6

Problem 2

Find the equation of a line with a slope of $-\frac{5}{2}$ and a point it passes through (10,-5).

Solution

Using the slope-intercept form, we can write the equation as:

y = $-\frac{5}{2}$ x + b

Substituting the given point (10,-5), we get:

-5 = $-\frac{5}{2}$ (10) + b

Simplifying the equation, we get:

-5 = -25 + b

Adding 25 to both sides, we get:

b = 20

The equation of the line is:

y = $-\frac{5}{2}$ x + 20

Applications

The slope-intercept form of a linear equation has many applications in mathematics and real-world problems. Some examples include:

Linear Regression: The slope-intercept form is used to model the relationship between two variables in linear regression.
Physics: The slope-intercept form is used to describe the motion of objects under constant acceleration.
Economics: The slope-intercept form is used to model the relationship between two variables in economics, such as the demand for a product.

Conclusion

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

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

Q: How do I find the equation of a line with a given slope and a point it passes through?

A: To find the equation of a line with a given slope and a point it passes through, you can use the slope-intercept form. First, substitute the given point into the equation to find the y-intercept. Then, use the slope and y-intercept to write the equation of the line.

Q: What is the y-intercept in the equation y = mx + b?

A: The y-intercept in the equation y = mx + b is the value of b. It represents the point where the line intersects the y-axis.

Q: How do I find the y-intercept in the equation y = mx + b?

A: To find the y-intercept in the equation y = mx + b, you can substitute the given point into the equation. For example, if the point is (x, y), you can substitute x and y into the equation and solve for b.

Q: What is the slope in the equation y = mx + b?

A: The slope in the equation y = mx + b is the value of m. It represents the rate of change of the line.

Q: How do I find the slope in the equation y = mx + b?

A: To find the slope in the equation y = mx + b, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: Can I use the slope-intercept form to find the equation of a line with a given slope and two points it passes through?

A: Yes, you can use the slope-intercept form to find the equation of a line with a given slope and two points it passes through. First, find the slope using the formula m = (y2 - y1) / (x2 - x1). Then, substitute one of the points into the equation to find the y-intercept. Finally, use the slope and y-intercept to write the equation of the line.

Q: What are some real-world applications of the slope-intercept form?

A: The slope-intercept form has many real-world applications, including:

Linear Regression: The slope-intercept form is used to model the relationship between two variables in linear regression.
Physics: The slope-intercept form is used to describe the motion of objects under constant acceleration.
Economics: The slope-intercept form is used to model the relationship between two variables in economics, such as the demand for a product.

Q: Can I use the slope-intercept form to find the equation of a line with a given slope and a point it passes through, if the point is not on the y-axis?

A: Yes, you can use the slope-intercept form to find the equation of a line with a given slope and a point it passes through, even if the point is not on the y-axis. First, substitute the given point into the equation to find the y-intercept. Then, use the slope and y-intercept to write the equation of the line.

Q: How do I know if a line is in slope-intercept form?

A: A line is in slope-intercept form if it can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: Can I use the slope-intercept form to find the equation of a line with a given slope and a point it passes through, if the slope is zero?

A: Yes, you can use the slope-intercept form to find the equation of a line with a given slope and a point it passes through, even if the slope is zero. In this case, the equation will be of the form y = b, where b is the y-intercept.

Q: How do I find the equation of a line with a given slope and a point it passes through, if the point is on the x-axis?

A: To find the equation of a line with a given slope and a point it passes through, if the point is on the x-axis, you can use the slope-intercept form. First, substitute the given point into the equation to find the y-intercept. Then, use the slope and y-intercept to write the equation of the line.