Finding the Equation of a Line that Goes Through a Given Point and Has a Known Slope

Oct 13, 2025 by windailys.com 85 views

Understanding the Basics of Linear Equations

In mathematics, a linear equation is a type of equation that represents a straight line on a coordinate plane. It is defined by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. 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 Equation of a Line in Slope-Intercept Form

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. This form is useful because it allows us to easily identify the slope and y-intercept of a line from its equation. For example, if we have the equation y = 2x + 3, we can see that the slope is 2 and the y-intercept is 3.

Finding the Equation of a Line that Goes Through a Given Point and Has a Known Slope

To find the equation of a line that goes through a given point and has a known slope, we can use the point-slope form of a linear equation. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.

Example: Finding the Equation of a Line that Goes Through the Point (0, -5) and Has a Slope of 3

Let's say we want to find the equation of a line that goes through the point (0, -5) and has a slope of 3. We can use the point-slope form of a linear equation to find the equation of the line.

Step 1: Write Down the Point-Slope Form of a Linear Equation

The point-slope form of a linear equation is y - y1 = m(x - x1). In this case, the given point is (0, -5) and the slope is 3. So, we can write down the equation as:

y - (-5) = 3(x - 0)

Step 2: Simplify the Equation

Now, we can simplify the equation by combining like terms. We can start by distributing the 3 to the terms inside the parentheses:

y + 5 = 3x

Step 3: Write the Equation in Slope-Intercept Form

Finally, we can write the equation in slope-intercept form by isolating the y-term on one side of the equation. We can do this by subtracting 5 from both sides of the equation:

y = 3x - 5

Conclusion

In conclusion, we have found the equation of a line that goes through the point (0, -5) and has a slope of 3. The equation is y = 3x - 5. This equation represents a straight line on a coordinate plane, and it can be used to graph the line and find its x- and y-intercepts.

Answer Key

The correct answer is A. y = 3x - 5.

Discussion

This problem is a classic example of how to find the equation of a line that goes through a given point and has a known slope. The point-slope form of a linear equation is a powerful tool that can be used to find the equation of a line in a variety of situations. By following the steps outlined above, we can find the equation of a line that meets a given set of conditions.

Practice Problems

Find the equation of a line that goes through the point (2, 3) and has a slope of 2.
Find the equation of a line that goes through the point (0, 2) and has a slope of -3.
Find the equation of a line that goes through the point (1, 4) and has a slope of 1.

Solutions

The equation of a line that goes through the point (2, 3) and has a slope of 2 is y = 2x + 1.
The equation of a line that goes through the point (0, 2) and has a slope of -3 is y = -3x + 2.
The equation of a line that goes through the point (1, 4) and has a slope of 1 is y = x + 3.

Q: What is a linear equation?

A: A linear equation is a type of equation that represents a straight line on a coordinate plane. It is defined by the equation y = mx + b, where m is the slope and b is the y-intercept.

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

A: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. This form is useful because it allows us to easily identify the slope and y-intercept of a line from its equation.

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

A: The point-slope form of a linear equation is y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. This form is useful because it allows us to find the equation of a line that goes through a given point and has a known slope.

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

A: To find the equation of a line that goes through a given point and has a known slope, we can use the point-slope form of a linear equation. We can start by writing down the point-slope form of the equation, and then simplify it to find the equation of the line.

Q: What is the equation of a line that goes through the point (0, -5) and has a slope of 3?

A: The equation of a line that goes through the point (0, -5) and has a slope of 3 is y = 3x - 5.

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

A: To find the x-intercept of a line, we can set y = 0 in the equation of the line and solve for x. This will give us the x-coordinate of the x-intercept.

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

A: To find the y-intercept of a line, we can set x = 0 in the equation of the line and solve for y. This will give us the y-coordinate of the y-intercept.

Q: What is the relationship between the slope and the y-intercept of a line?

A: The slope and the y-intercept of a line are related by the equation y = mx + b, where m is the slope and b is the y-intercept. This equation shows that the slope and y-intercept are both important components of a linear equation.

Q: Can a line have a slope of 0?

A: Yes, a line can have a slope of 0. This occurs when the line is horizontal, and the equation of the line is of the form y = b.

Q: Can a line have a slope of infinity?

A: No, a line cannot have a slope of infinity. This occurs when the line is vertical, and the equation of the line is of the form x = a.

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

A: To graph a line on a coordinate plane, we can use the equation of the line to find the x- and y-intercepts of the line. We can then plot these points on the coordinate plane and draw a line through them to represent the graph of the line.

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

A: To find the equation of a line that goes through the points (1, 2) and (3, 4), we can use the point-slope form of a linear equation. We can start by writing down the point-slope form of the equation, and then simplify it to find the equation of the line.

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

A: To find the equation of a line that goes through two points, we can use the point-slope form of a linear equation. We can start by writing down the point-slope form of the equation, and then simplify it to find the equation of the line.

Q: What is the relationship between the slope and the x-intercept of a line?

A: The slope and the x-intercept of a line are related by the equation y = mx + b, where m is the slope and b is the y-intercept. This equation shows that the slope and x-intercept are both important components of a linear equation.

Q: Can a line have a y-intercept of 0?

A: Yes, a line can have a y-intercept of 0. This occurs when the line is horizontal, and the equation of the line is of the form y = 0.

Q: Can a line have a y-intercept of infinity?

A: No, a line cannot have a y-intercept of infinity. This occurs when the line is vertical, and the equation of the line is of the form x = a.

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

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

A: The equation of a line that goes through the point (2, 3) and has a slope of 2 is y = 2x + 1.

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

Q: What is the relationship between the slope and the y-intercept of a line?

Q: Can a line have a slope of -1?

A: Yes, a line can have a slope of -1. This occurs when the line is a reflection of the x-axis, and the equation of the line is of the form y = -x + b.

Q: Can a line have a slope of 1?

A: Yes, a line can have a slope of 1. This occurs when the line is a reflection of the y-axis, and the equation of the line is of the form y = x + b.

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

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

A: The equation of a line that goes through the point (1, 2) and has a slope of 2 is y = 2x + 1.

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

Q: What is the relationship between the slope and the x-intercept of a line?

Q: Can a line have a y-intercept of -1?

A: Yes, a line can have a y-intercept of -1. This occurs when the line is a reflection of the x-axis, and the equation of the line is of the form y = -x - 1.

Q: Can a line have a y-intercept of 1?

A: Yes, a line can have a y-intercept of 1. This occurs when the line is a reflection of the y-axis, and the equation of the line is of the form y = x + 1.