Writing equations of lines

Oct 13, 2025 by windailys.com 58 views

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Introduction

In mathematics, the point-slope form of a linear equation is a powerful tool for describing lines on a coordinate plane. Given a point on the line and its slope, we can use the point-slope form to write the equation of the line. In this article, we will explore how to find the equation of a line with a slope of -0.2 that passes through the point (11, 5) using the point-slope form.

The Point-Slope Form of a Linear Equation

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

y - y1 = m(x - x1)

where (x1, y1) is a point on the line, and m is the slope of the line. This form is useful because it allows us to easily write the equation of a line given a point on the line and its slope.

Finding the Equation of the Line

Now that we have the point-slope form, we can use it to find the equation of the line with a slope of -0.2 that passes through the point (11, 5). We will substitute the values of the point and the slope into the point-slope form and simplify the resulting equation.

Step 1: Substitute the Values of the Point and the Slope

We are given that the slope of the line is -0.2 and that the point (11, 5) lies on the line. We will substitute these values into the point-slope form:

y - 5 = -0.2(x - 11)

Step 2: Simplify the Equation

Now that we have substituted the values, we can simplify the equation by distributing the slope to the terms inside the parentheses:

y - 5 = -0.2x + 2.2

Step 3: Write the Equation in Standard Form

To write the equation in standard form, we will add 5 to both sides of the equation:

y = -0.2x + 7.2

Conclusion

In this article, we used the point-slope form to find the equation of a line with a slope of -0.2 that passes through the point (11, 5). We substituted the values of the point and the slope into the point-slope form, simplified the resulting equation, and wrote the equation in standard form. The equation of the line is y = -0.2x + 7.2.

Example Problems

Problem 1

Find the equation of a line with a slope of 0.5 that passes through the point (3, 2).

Solution

We will use the point-slope form to find the equation of the line:

y - 2 = 0.5(x - 3)

Simplifying the equation, we get:

y - 2 = 0.5x - 1.5

Adding 2 to both sides, we get:

y = 0.5x + 0.5

Problem 2

Find the equation of a line with a slope of -1 that passes through the point (4, 3).

Solution

We will use the point-slope form to find the equation of the line:

y - 3 = -1(x - 4)

Simplifying the equation, we get:

y - 3 = -x + 4

Adding 3 to both sides, we get:

y = -x + 7

Applications of the Point-Slope Form

The point-slope form is a powerful tool for describing lines on a coordinate plane. It is used in a variety of applications, including:

Graphing lines: The point-slope form is used to graph lines on a coordinate plane.
Writing equations of lines: The point-slope form is used to write the equation of a line given a point on the line and its slope.
Solving systems of equations: The point-slope form is used to solve systems of linear equations.
Modeling real-world problems: The point-slope form is used to model real-world problems, such as the motion of an object or the growth of a population.

Conclusion

In conclusion, the point-slope form is a powerful tool for describing lines on a coordinate plane. It is used to write the equation of a line given a point on the line and its slope. In this article, we used the point-slope form to find the equation of a line with a slope of -0.2 that passes through the point (11, 5). We also explored some example problems and applications of the point-slope form.

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Introduction

In our previous article, we explored how to find the equation of a line with a slope of -0.2 that passes through the point (11, 5) using the point-slope form. In this article, we will answer some frequently asked questions about the point-slope form and its applications.

Q&A

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

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

y - y1 = m(x - x1)

where (x1, y1) 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 a point on the line and its slope?

A: To find the equation of a line given a point on the line and its slope, you can use the point-slope form. Simply substitute the values of the point and the slope into the point-slope form and simplify the resulting equation.

Q: What is the difference between the point-slope form and the standard form of a linear equation?

A: The point-slope form and the standard form of a linear equation are two different ways of writing the equation of a line. The point-slope form is given by:

y - y1 = m(x - x1)

while the standard form is given by:

Ax + By = C

where A, B, and C are constants.

Q: Can I use the point-slope form to find the equation of a line with a slope of 0?

A: Yes, you can use the point-slope form to find the equation of a line with a slope of 0. In this case, the equation will be of the form:

y = y1

Q: How do I use the point-slope form to solve systems of linear equations?

A: To use the point-slope form to solve systems of linear equations, you can substitute the equations into the point-slope form and solve for the variables.

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

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

Graphing lines: The point-slope form is used to graph lines on a coordinate plane.
Writing equations of lines: The point-slope form is used to write the equation of a line given a point on the line and its slope.
Solving systems of equations: The point-slope form is used to solve systems of linear equations.
Modeling real-world problems: The point-slope form is used to model real-world problems, such as the motion of an object or the growth of a population.

Example Problems

Problem 1

Find the equation of a line with a slope of 0.5 that passes through the point (3, 2).

Solution

We will use the point-slope form to find the equation of the line:

y - 2 = 0.5(x - 3)

Simplifying the equation, we get:

y - 2 = 0.5x - 1.5

Adding 2 to both sides, we get:

y = 0.5x + 0.5

Problem 2

Find the equation of a line with a slope of -1 that passes through the point (4, 3).

Solution

We will use the point-slope form to find the equation of the line:

y - 3 = -1(x - 4)

Simplifying the equation, we get:

y - 3 = -x + 4

Adding 3 to both sides, we get:

y = -x + 7

Conclusion

In conclusion, the point-slope form is a powerful tool for describing lines on a coordinate plane. It is used to write the equation of a line given a point on the line and its slope. In this article, we answered some frequently asked questions about the point-slope form and its applications. We also explored some example problems and real-world applications of the point-slope form.

Additional Resources

Graphing lines: The point-slope form is used to graph lines on a coordinate plane.
Writing equations of lines: The point-slope form is used to write the equation of a line given a point on the line and its slope.
Solving systems of equations: The point-slope form is used to solve systems of linear equations.
Modeling real-world problems: The point-slope form is used to model real-world problems, such as the motion of an object or the growth of a population.

Final Thoughts

The point-slope form is a fundamental concept in mathematics, and it has many real-world applications. It is used to write the equation of a line given a point on the line and its slope. In this article, we explored some frequently asked questions about the point-slope form and its applications. We also explored some example problems and real-world applications of the point-slope form.