Which Equation Represents A Line That Passes Through { \left(2, -\frac{1}{2}\right)$}$ And Has A Slope Of 3?A. { Y + \frac{1}{2} = 3(x - 2)$}$B. { Y - 3 = 2\left(x + \frac{1}{2}\right)$} C . \[ C. \[ C . \[ Y + \frac{1}{2} = 2(x -

Mar 13, 2025 by ADMIN 231 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 is used to describe the relationship between two variables. A linear equation can be written in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.

The Slope-Intercept Form of a Linear Equation

The slope-intercept form of a linear equation is a way of writing an equation in the form y = mx + b, where m is the slope and b is the y-intercept. The slope of a line is a measure of how steep it is, and is calculated as the ratio of the vertical change (rise) to the horizontal change (run). The y-intercept is the point at which the line intersects the y-axis.

The Problem at Hand

We are given a point (2, -1/2) and a slope of 3, and 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.

Using the Point-Slope Form to Find the Equation

Using the point-slope form, we can plug in the values of the given point and the slope to get:

y - (-1/2) = 3(x - 2)

Simplifying this equation, we get:

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

Comparing the Derived Equation with the Options

Now that we have derived the equation of the line that passes through the given point and has the given slope, we can compare it with the options given in the problem. We have:

A. y + 1/2 = 3(x - 2) B. y - 3 = 2(x + 1/2) C. y + 1/2 = 2(x - 2)

Comparing the derived equation with the options, we can see that option A is the correct answer.

Conclusion

In this article, we have discussed the basics of linear equations and the slope-intercept form of a linear equation. We have also used the point-slope form to find the equation of a line that passes through a given point and has a specific slope. We have compared the derived equation with the options given in the problem and have found that option A is the correct answer.

Key Takeaways

A linear equation is a type of equation that represents a line on a coordinate plane.
The slope-intercept form of a linear equation is a way of writing an equation in the form y = mx + b, where m is the slope and b is the y-intercept.
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.
To find the equation of a line that passes through a given point and has a specific slope, we can use the point-slope form.

Frequently Asked Questions

What is the slope-intercept form of a linear equation?
What is the point-slope form of a linear equation?
How do we find the equation of a line that passes through a given point and has a specific slope?

Answer to Frequently Asked Questions

The slope-intercept form of a linear equation is a way of writing an equation in the form y = mx + b, where m is the slope and b is the y-intercept.
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.
To find the equation of a line that passes through a given point and has a specific slope, we can use the point-slope form.

References

[1] "Linear Equations" by Math Open Reference
[2] "Slope-Intercept Form" by Khan Academy
[3] "Point-Slope Form" by Purplemath
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 is used to describe the relationship between two variables.

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

A: The slope-intercept form of a linear equation is a way of writing an equation in the form y = mx + b, where m is the slope and b is the y-intercept. The slope of a line is a measure of how steep it is, and is calculated as the ratio of the vertical change (rise) to the horizontal change (run).

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. This form is useful for finding the equation of a line that passes through a given point and has a specific 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 plug in the values of the given point and the slope into the equation, and simplify to get the final equation.

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

A: The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to 0.

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

A: To find the y-intercept of a line, you can use the slope-intercept form of a linear equation. Simply plug in the values of the slope and the y-intercept into the equation, and solve for y.

Q: What is the slope of a line?

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

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

A: To find 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 equation of a line that passes through the points (2, 3) and (4, 5)?

A: To find the equation of a line that passes through the points (2, 3) and (4, 5), you can use the point-slope form of a linear equation. Simply plug in the values of the points and the slope into the equation, and simplify to get the final equation.

Q: What is the equation of a line that has a slope of 2 and passes through the point (1, 3)?

A: To find the equation of a line that has a slope of 2 and passes through the point (1, 3), you can use the point-slope form of a linear equation. Simply plug in the values of the point and the slope into the equation, and simplify to get the final equation.

Q: How do I graph a line on a coordinate plane?

A: To graph a line on a coordinate plane, you can use the slope-intercept form of a linear equation. Simply plot the y-intercept on the y-axis, and then use the slope to find the next point on the line. Continue plotting points until you have a complete graph of the line.

Q: What is the equation of a line that is parallel to the line y = 2x + 3?

A: To find the equation of a line that is parallel to the line y = 2x + 3, you can use the fact that parallel lines have the same slope. Simply write 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 equation of a line that is perpendicular to the line y = 2x + 3?

A: To find the equation of a line that is perpendicular to the line y = 2x + 3, you can use the fact that perpendicular lines have slopes that are negative reciprocals of each other. Simply write the equation of the line in the form 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. Simply plug in the values of the points and the slope into the equation, and simplify to get the final equation.

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

A: To find the equation of a line that passes through the points (1, 2) and (3, 4), you can use the point-slope form of a linear equation. Simply plug in the values of the points and the slope into the equation, and simplify to get the final equation.

Q: How do I find the equation of a line that has a slope of 3 and passes through the point (2, 1)?

A: To find the equation of a line that has a slope of 3 and passes through the point (2, 1), you can use the point-slope form of a linear equation. Simply plug in the values of the point and the slope into the equation, and simplify to get the final equation.

Q: What is the equation of a line that is horizontal?

A: The equation of a line that is horizontal is y = b, where b is the y-intercept.

Q: What is the equation of a line that is vertical?

A: The equation of a line that is vertical is x = a, where a is the x-intercept.

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 plug in the values of the point and the slope into the equation, and simplify to get the final equation.

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

Q: How do I find the equation of a line that has a slope of 2 and passes through the point (1, 3)?

Q: What is the equation of a line that is parallel to the line y = 2x + 3?

Q: What is the equation of a line that is perpendicular to the line y = 2x + 3?

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

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

Q: How do I find the equation of a line that has a slope of 3 and passes through the point (2, 1)?

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