Checking that Both Points Satisfy the Equation

Oct 13, 2025 by pynkocean.com 46 views

Introduction

In mathematics, the equation of a line can be found using various methods, including the slope-intercept form. This form is represented by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will discuss how to find the equation of a line through two given points using the slope-intercept form.

Step 1: Find the Slope of the Line

To find the equation of a line through two points, we first need to find the slope of the line. The slope of a line is a measure of how steep it is and can be calculated using the formula:

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

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Let's use the given points (5, -2) and (1, 4) to find the slope of the line.

Calculating the Slope

m = (4 - (-2)) / (1 - 5) m = (4 + 2) / (-4) m = 6 / (-4) m = -3/2

So, the slope of the line is -3/2.

Step 2: Find the Equation of the Line

Now that we have the slope of the line, we can use the slope-intercept form to find the equation of the line. We can use either of the two points to substitute into the equation and solve for b.

Let's use the point (5, -2) to substitute into the equation.

Substituting the Point into the Equation

y = mx + b -2 = (-3/2)(5) + b -2 = (-15/2) + b -2 + 15/2 = b -4/2 + 15/2 = b 11/2 = b

So, the equation of the line is y = (-3/2)x + 11/2.

Checking that Both Points Satisfy the Equation

To verify that the equation of the line is correct, we need to check that both points satisfy the equation.

Checking the First Point

y = (-3/2)x + 11/2 -2 = (-3/2)(5) + 11/2 -2 = (-15/2) + 11/2 -2 = (-4/2) -2 = -2

The first point (5, -2) satisfies the equation.

Checking the Second Point

y = (-3/2)x + 11/2 4 = (-3/2)(1) + 11/2 4 = (-3/2) + 11/2 4 = (8/2) 4 = 4

The second point (1, 4) also satisfies the equation.

Conclusion

In this article, we discussed how to find the equation of a line through two given points using the slope-intercept form. We found the slope of the line using the formula m = (y2 - y1) / (x2 - x1) and then used the slope-intercept form to find the equation of the line. We also checked that both points satisfy the equation to verify that the equation is correct.

Key Takeaways

The slope of a line can be calculated using the formula m = (y2 - y1) / (x2 - x1).
The equation of a line can be found using the slope-intercept form y = mx + b.
To verify that the equation of a line is correct, we need to check that both points satisfy the equation.

References

[1] Khan Academy. (n.d.). Slope-intercept form. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f0/x2f1f0/slope-intercept-form
[2] Math Open Reference. (n.d.). Slope-intercept form. Retrieved from https://www.mathopenref.com/slopeintercept.html

Glossary

Slope: A measure of how steep a line is.
Slope-intercept form: A way of writing the equation of a line in the form y = mx + b.
Y-intercept: The point where the line intersects the y-axis.
Frequently Asked Questions about Finding the Equation of a Line ====================================================================

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

A: The slope-intercept form of a line is a way of writing the equation of a line in the form y = mx + b, where m is the slope of the line and b is the y-intercept.

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 the coordinates of 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 intersects the y-axis. It is the value of y when x is equal to 0.

Q: How do I find the equation of a line if I know the slope and a point on the line?

A: To find the equation of a line if you know the slope and a point on the line, you can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope.

Q: Can I use any two points to find the equation of a line?

A: Yes, you can use any two points to find the equation of a line. However, if the two points are the same, the line will be a single point and will not have a slope.

Q: How do I check if a point is on a line?

A: To check if a point is on a line, you can substitute the coordinates of the point into the equation of the line and see if it is true.

Q: What if I have a line with a negative slope?

A: If you have a line with a negative slope, it means that the line slopes downward from left to right. This is the opposite of a line with a positive slope, which slopes upward from left to right.

Q: Can I use the slope-intercept form to find the equation of a horizontal line?

A: Yes, you can use the slope-intercept form to find the equation of a horizontal line. A horizontal line has a slope of 0, so the equation will be in the form y = b.

Q: Can I use the slope-intercept form to find the equation of a vertical line?

A: No, you cannot use the slope-intercept form to find the equation of a vertical line. A vertical line has an undefined slope, so the equation will be in the form x = a.

Q: What if I have a line with a slope of 1 or -1?

A: If you have a line with a slope of 1 or -1, it means that the line slopes upward or downward at a 45-degree angle. This is a special case, and the equation of the line will be in the form y = x + b or y = -x + b.

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

A: Yes, you can use the slope-intercept form to find the equation of a line with a fractional slope. For example, if the slope is 3/4, the equation will be in the form y = (3/4)x + b.

Conclusion

In this article, we have answered some frequently asked questions about finding the equation of a line. We have discussed the slope-intercept form, how to find the slope of a line, and how to check if a point is on a line. We have also covered some special cases, such as horizontal and vertical lines, and lines with fractional slopes.