Which Equation Represents A Line That Passes Through { (5,1)$}$ And Has A Slope Of { \frac{1}{2}$}$?A. { Y - 5 = \frac{1}{2}(x - 1)$}$B. { Y - \frac{1}{2} = 5(x - 1)$} C . \[ C. \[ C . \[ Y - 1 = \frac{1}{2}(x -

Mar 7, 2025 by ADMIN 212 views

Which Equation Represents a Line That Passes Through a Given Point and Has a Specific Slope?

Understanding the Basics of Linear Equations

In mathematics, a linear equation is a type of equation that represents a line on a coordinate plane. It is a fundamental concept in algebra and geometry, and it has numerous applications in various fields, including physics, engineering, and economics. A linear equation is typically represented in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.

The Importance of Slope and Point in Linear Equations

The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope is denoted by the letter m, and it can be positive, negative, or zero. A positive slope indicates that the line slopes upward from left to right, while a negative slope indicates that the line slopes downward from left to right. A slope of zero indicates that the line is horizontal.

In addition to the slope, a linear equation also requires a point through which the line passes. This point is called the x-intercept or the y-intercept, depending on whether it lies on the x-axis or the y-axis. The x-intercept is the point where the line crosses the x-axis, and it has a y-coordinate of zero. The y-intercept is the point where the line crosses the y-axis, and it has an x-coordinate of zero.

The Problem at Hand

In this problem, we are given a point (5,1) and a slope of 1/2. We need to find the equation of the line that passes through this point and has this slope. To do this, we can use the point-slope form of a linear equation, which is given by:

y - y1 = m(x - x1)

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

Analyzing the Options

Let's analyze the options given in the problem:

A. y - 5 = 1/2(x - 1)

B. y - 1/2 = 5(x - 1)

C. y - 1 = 1/2(x - 5)

We need to determine which of these options represents the line that passes through the point (5,1) and has a slope of 1/2.

Option A

Option A is y - 5 = 1/2(x - 1). To determine whether this option is correct, we can substitute the point (5,1) into the equation and check if it satisfies the equation.

y - 5 = 1/2(x - 1) 1 - 5 = 1/2(5 - 1) -4 = 1/2(4) -4 = 2

This is not true, so option A is not correct.

Option B

Option B is y - 1/2 = 5(x - 1). To determine whether this option is correct, we can substitute the point (5,1) into the equation and check if it satisfies the equation.

y - 1/2 = 5(x - 1) 1 - 1/2 = 5(5 - 1) 1/2 = 5(4) 1/2 = 20

This is not true, so option B is not correct.

Option C

Option C is y - 1 = 1/2(x - 5). To determine whether this option is correct, we can substitute the point (5,1) into the equation and check if it satisfies the equation.

y - 1 = 1/2(x - 5) 1 - 1 = 1/2(5 - 5) 0 = 1/2(0) 0 = 0

This is true, so option C is correct.

Conclusion

Understanding 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. This form is useful for finding the equation of a line that passes through a given point and has a specific slope.

Real-World Applications

Linear equations have numerous real-world applications, including:

Physics: Linear equations are used to describe the motion of objects under constant acceleration.
Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Linear equations are used to model economic systems and make predictions about future trends.

Final Thoughts

In conclusion, the equation that represents a line that passes through the point (5,1) and has a slope of 1/2 is y - 1 = 1/2(x - 5). This equation is in the point-slope form, and it satisfies the given conditions. Linear equations have numerous real-world applications, and they are an essential tool for problem-solving in mathematics and other fields.
Frequently Asked Questions About Linear Equations

Q: What is a linear equation?

A: A linear equation is a type of equation that represents a line on a coordinate plane. It is a fundamental concept in algebra and geometry, and it has numerous applications in various fields, including physics, engineering, and economics.

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 the given point and m is the slope.

Q: How do I find the equation of a line that passes through a given point and has a specific slope?

A: To find the equation of a line that passes through a given point and has a specific slope, you can use the point-slope form of a linear equation. Simply substitute the given point and slope into the equation, and simplify to find the equation of the line.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I calculate the slope of a line?

A: To calculate the slope of a line, you can use the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line.

Q: What is the y-intercept of a line?

A: The y-intercept of a line is the point where the line crosses the y-axis. It has an x-coordinate of zero and a y-coordinate that is equal to the y-intercept.

Q: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, you can use the equation of the line in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the x-intercept of a line?

A: The x-intercept of a line is the point where the line crosses the x-axis. It has a y-coordinate of zero and an x-coordinate that is equal to the x-intercept.

Q: How do I find the x-intercept of a line?

A: To find the x-intercept of a line, you can set the y-coordinate equal to zero and solve for the x-coordinate.

Q: Can I use linear equations to solve real-world problems?

A: Yes, linear equations have numerous real-world applications, including physics, engineering, and economics. They can be used to model and solve problems in a wide range of fields.

Q: What are some common applications of linear equations?

A: Some common applications of linear equations include:

Physics: Linear equations are used to describe the motion of objects under constant acceleration.
Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Linear equations are used to model economic systems and make predictions about future trends.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can use the equation in the form y = mx + b, where m is the slope and b is the y-intercept. You can also use the point-slope form of a linear equation to graph the line.

Q: What are some common mistakes to avoid when working with linear equations?

A: Some common mistakes to avoid when working with linear equations include:

Not using the correct form of the equation (e.g. using the slope-intercept form instead of the point-slope form).
Not substituting the correct values into the equation.
Not simplifying the equation correctly.
Not checking the solution for errors.

Q: How do I check my work when solving linear equations?

A: To check your work when solving linear equations, you can:

Plug in the solution into the original equation to see if it is true.
Graph the equation to see if the solution is correct.
Use a calculator to check the solution.
Check the solution for errors, such as division by zero or negative numbers.