Finding equations of lines

Oct 13, 2025 by windailys.com 19 views

Introduction

In mathematics, the point-slope form is a method used to write the equation of a line that passes through a given point and has a specified slope. This form is particularly useful when we are given the coordinates of a point on the line and the slope of the line. In this article, we will explore how to use the point-slope form to write the equation of a line that passes through the point $$(7,19)$$ with slope $$\frac{14}{9}$$.

What is Point-Slope Form?

The point-slope form of a linear equation is given by the formula:

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. This formula allows us to write the equation of a line that passes through a given point and has a specified slope.

Step 1: Identify the Given Point and Slope

In this problem, we are given the point $$(7,19)$$ and the slope $$\frac{14}{9}$$. We can use these values to write the equation of the line using the point-slope form.

Step 2: Plug in the Values into the Point-Slope Form

To write the equation of the line, we need to plug in the values of the given point and slope into the point-slope form. We can do this by substituting $$(x_1, y_1) = (7,19)$$ and $$m = \frac{14}{9}$$ into the formula:

y - 19 = \frac{14}{9}(x - 7)

Step 3: Simplify the Equation

To simplify the equation, we can start by multiplying both sides of the equation by 9 to eliminate the fraction:

9(y - 19) = 14(x - 7)

Next, we can distribute the 9 to the terms inside the parentheses:

9y - 171 = 14x - 98

Step 4: Write the Equation in Standard Form

To write the equation in standard form, we need to isolate the constant term on one side of the equation. We can do this by adding 171 to both sides of the equation:

9y = 14x + 73

Conclusion

In this article, we used the point-slope form to write the equation of a line that passes through the point $$(7,19)$$ with slope $$\frac{14}{9}$$. We started by identifying the given point and slope, then plugged in the values into the point-slope form. We simplified the equation by multiplying both sides by 9 and distributing the 9 to the terms inside the parentheses. Finally, we wrote the equation in standard form by isolating the constant term on one side of the equation.

Example Problems

Problem 1

Use the point-slope form to write the equation of a line that passes through the point $$(3,5)$$ with slope $$\frac{2}{3}$$.

Solution

To write the equation of the line, we need to plug in the values of the given point and slope into the point-slope form:

y - 5 = \frac{2}{3}(x - 3)

Next, we can simplify the equation by multiplying both sides by 3 to eliminate the fraction:

3(y - 5) = 2(x - 3)

Distributing the 3 to the terms inside the parentheses, we get:

3y - 15 = 2x - 6

Adding 15 to both sides of the equation, we get:

3y = 2x + 9

Problem 2

Use the point-slope form to write the equation of a line that passes through the point $$(2,7)$$ with slope $$\frac{3}{4}$$.

Solution

To write the equation of the line, we need to plug in the values of the given point and slope into the point-slope form:

y - 7 = \frac{3}{4}(x - 2)

Next, we can simplify the equation by multiplying both sides by 4 to eliminate the fraction:

4(y - 7) = 3(x - 2)

Distributing the 4 to the terms inside the parentheses, we get:

4y - 28 = 3x - 6

Adding 28 to both sides of the equation, we get:

4y = 3x + 22

Applications of Point-Slope Form

The point-slope form is a powerful tool for writing the equation of a line that passes through a given point and has a specified slope. This form is particularly useful in a variety of applications, including:

Graphing lines: The point-slope form can be used to graph lines by plotting the given point and using the slope to determine the direction of the line.
Finding equations of lines: The point-slope form can be used to find the equation of a line that passes through a given point and has a specified slope.
Solving systems of equations: The point-slope form can be used to solve systems of equations by substituting the equation of one line into the equation of another line.

Conclusion

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

A: The point-slope form of a linear equation is given by the formula:

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 use the point-slope form to write the equation of a line?

A: To use the point-slope form to write the equation of a line, you need to:

Identify the given point and slope.
Plug in the values of the given point and slope into the point-slope form.
Simplify the equation by multiplying both sides by the denominator of the slope.
Distribute the denominator to the terms inside the parentheses.
Write the equation in standard form by isolating the constant term on one side of the equation.

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

A: The point-slope form and the slope-intercept form are two different ways to write the equation of a line. The point-slope form is given by the formula:

y - y_1 = m(x - x_1)

while the slope-intercept form is given by the formula:

y = mx + b

The point-slope form is useful when you are given the coordinates of a point on the line and the slope of the line, while the slope-intercept form is useful when you are given the slope of the line and the y-intercept.

Q: Can I use the point-slope form to write the equation of a line that is not in the standard form?

A: Yes, you can use the point-slope form to write the equation of a line that is not in the standard form. However, you need to be careful when simplifying the equation to ensure that it is in the standard form.

Q: How do I find the slope of a line using the point-slope form?

A: To find the slope of a line using the point-slope form, you need to:

Identify the given point and the equation of the line.
Plug in the values of the given point into the equation of the line.
Simplify the equation to isolate the slope.
The slope is the coefficient of the x-term.

Q: Can I use the point-slope form to write the equation of a line that is not a straight line?

A: No, the point-slope form is only used to write the equation of a straight line. If you are given a curve or a non-linear equation, you need to use a different method to write the equation.

Q: How do I graph a line using the point-slope form?

A: To graph a line using the point-slope form, you need to:

Identify the given point and the slope of the line.
Plot the given point on the coordinate plane.
Use the slope to determine the direction of the line.
Draw a line that passes through the given point and has the specified slope.

Q: Can I use the point-slope form to write the equation of a line that is not in the slope-intercept form?

A: Yes, you can use the point-slope form to write the equation of a line that is not in the slope-intercept form. However, you need to be careful when simplifying the equation to ensure that it is in the standard form.

Conclusion

In this article, we answered some frequently asked questions about the point-slope form. We covered topics such as how to use the point-slope form to write the equation of a line, the difference between the point-slope form and the slope-intercept form, and how to graph a line using the point-slope form. We also covered some common mistakes to avoid when using the point-slope form.