Understanding Slope and Point

Oct 13, 2025 by miodeya.com 82 views

Introduction

In mathematics, linear equations are a fundamental concept that describe a relationship between two variables. The equation of a line can be expressed in various forms, including slope-intercept form, point-slope form, and standard form. In this article, we will focus on finding the equation of a line that passes through a given point and has a specific slope.

Understanding Slope and Point

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of a line can be represented by the letter 'm'. In this case, we are given that the slope of the line is 1.

The point through which the line passes is given as (1,5). This means that the line passes through the x-coordinate of 1 and the y-coordinate of 5.

Using the Point-Slope Form

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

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope.

In this case, we are given the point (1,5) and the slope (m) is 1. Plugging these values into the point-slope form, we get:

y - 5 = 1(x - 1)

Simplifying the Equation

To simplify the equation, we can start by distributing the slope (1) to the terms inside the parentheses:

y - 5 = x - 1

Next, we can add 5 to both sides of the equation to isolate the y-term:

y = x + 4

Comparing with the Options

Now that we have the equation of the line, we can compare it with the options given:

A. y - 1 = (x + 5) B. y - 5 = (x - 1) C. y + 1 = (x - 5) D. y + 5 = (x + 1)

Comparing our equation (y = x + 4) with the options, we can see that none of the options exactly match our equation. However, we can rewrite our equation in a form that matches one of the options.

Rewriting the Equation

We can rewrite our equation (y = x + 4) in the form y - 5 = (x - 1) by subtracting 5 from both sides:

y - 5 = x - 1

This is option B.

Conclusion

In conclusion, the equation of the line that has a slope of 1 and passes through the point (1,5) is y - 5 = (x - 1). This is option B.

Key Takeaways

The slope of a line is a measure of how steep it is.
The point-slope form of a linear equation is given by y - y1 = m(x - x1).
To find the equation of a line, we can use the point-slope form and plug in the given point and slope.
We can simplify the equation by distributing the slope and adding or subtracting terms to isolate the y-term.
We can compare the equation with the options given to find the correct answer.

Practice Problems

Find the equation of the line that has a slope of 2 and passes through the point (3,7).
Find the equation of the line that has a slope of -3 and passes through the point (2,9).
Find the equation of the line that has a slope of 1 and passes through the point (4,2).

Answer Key

y - 7 = 2(x - 3)
y - 9 = -3(x - 2)
y - 2 = (x - 4)
Solving Linear Equations: Q&A =============================

Introduction

In our previous article, we discussed how to find the equation of a line that passes through a given point and has a specific slope. In this article, we will answer some frequently asked questions related to solving linear equations.

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 and b is the y-intercept.

Q: How do I find the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, you can use the point-slope form of a linear equation. First, find the slope of the line using the formula m = (y2 - y1) / (x2 - x1). Then, plug in one of the points and the slope into the point-slope form to get the equation.

Q: What is the standard form of a linear equation?

A: The standard form of a linear equation is given by Ax + By = C, where A, B, and C are constants.

Q: How do I convert a linear equation from slope-intercept form to standard form?

A: To convert a linear equation from slope-intercept form to standard form, you can multiply both sides of the equation by the denominator of the slope (if it is a fraction) and then rearrange the terms to get the equation in the form Ax + By = C.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable (x) is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable (x) is 2.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use the method of substitution or elimination. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Q: What is the equation of the line that passes through the points (2,3) and (4,5)?

A: To find the equation of the line that passes through the points (2,3) and (4,5), we can use the point-slope form of a linear equation. First, find the slope of the line using the formula m = (y2 - y1) / (x2 - x1). Plugging in the values, we get m = (5 - 3) / (4 - 2) = 2 / 2 = 1. Then, plug in one of the points and the slope into the point-slope form to get the equation.

y - 3 = 1(x - 2)

Simplifying the equation, we get:

y - 3 = x - 2

Adding 3 to both sides, we get:

y = x + 1

Q: What is the equation of the line that passes through the points (1,2) and (3,4)?

A: To find the equation of the line that passes through the points (1,2) and (3,4), we can use the point-slope form of a linear equation. First, find the slope of the line using the formula m = (y2 - y1) / (x2 - x1). Plugging in the values, we get m = (4 - 2) / (3 - 1) = 2 / 2 = 1. Then, plug in one of the points and the slope into the point-slope form to get the equation.

y - 2 = 1(x - 1)

Simplifying the equation, we get:

y - 2 = x - 1

Adding 2 to both sides, we get:

y = x + 1

Conclusion

In conclusion, solving linear equations is an important concept in mathematics that has many real-world applications. By understanding the different forms of linear equations and how to solve them, you can apply this knowledge to a wide range of problems.

Practice Problems

Find the equation of the line that passes through the points (2,3) and (4,5).
Find the equation of the line that passes through the points (1,2) and (3,4).
Find the equation of the line that has a slope of 2 and passes through the point (3,7).

Answer Key

y = x + 1
y = x + 1
y - 7 = 2(x - 3)