Solve For $y_1 - Y_2$:$m = \frac{y_1 - Y_2}{x_1 - X_2}$

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Introduction

In mathematics, the concept of slope is a fundamental idea in algebra and geometry. The slope of a line is a measure of how steep it is, and it can be calculated using the formula: $m = \frac{y_1 - y_2}{x_1 - x_2}$. In this article, we will focus on solving for the difference in y-values, denoted as $y_1 - y_2$, given the slope and the coordinates of two points on the line.

Understanding the Formula

The formula for slope is: $m = \frac{y_1 - y_2}{x_1 - x_2}$. This formula calculates the slope of a line by dividing the difference in y-values by the difference in x-values. To solve for the difference in y-values, we need to isolate $y_1 - y_2$ on one side of the equation.

Solving for the Difference in y-Values

To solve for the difference in y-values, we can start by multiplying both sides of the equation by $x_1 - x_2$. This will eliminate the fraction and give us: $m(x_1 - x_2) = y_1 - y_2$.

Next, we can add $y_2$ to both sides of the equation to get: $m(x_1 - x_2) + y_2 = y_1$.

Now, we can subtract $y_2$ from both sides of the equation to get: $m(x_1 - x_2) = y_1 - y_2$.

Finally, we can divide both sides of the equation by $m$ to get: $\frac{m(x_1 - x_2)}{m} = \frac{y_1 - y_2}{m}$.

Simplifying the equation, we get: $x_1 - x_2 = \frac{y_1 - y_2}{m}$.

Multiplying Both Sides by m

Multiplying both sides of the equation by $m$, we get: $m(x_1 - x_2) = y_1 - y_2$.

Adding y2 to Both Sides

Adding $y_2$ to both sides of the equation, we get: $m(x_1 - x_2) + y_2 = y_1$.

Subtracting y2 from Both Sides

Subtracting $y_2$ from both sides of the equation, we get: $m(x_1 - x_2) = y_1 - y_2$.

Dividing Both Sides by m

Dividing both sides of the equation by $m$, we get: $\frac{m(x_1 - x_2)}{m} = \frac{y_1 - y_2}{m}$.

Simplifying the Equation

Simplifying the equation, we get: $x_1 - x_2 = \frac{y_1 - y_2}{m}$.

Multiplying Both Sides by m

Multiplying both sides of the equation by $m$, we get: $m(x_1 - x_2) = y_1 - y_2$.

The Final Answer

The final answer is: $y_1 - y_2 = m(x_1 - x_2)$.

Example Problem

Let's say we have two points on a line: $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (4, 5)$. The slope of the line is $m = 2$. We want to find the difference in y-values, denoted as $y_1 - y_2$.

Using the formula, we can plug in the values: $m = 2$, $x_1 = 2$, $x_2 = 4$, $y_1 = 3$, and $y_2 = 5$.

Substituting these values into the formula, we get: $y_1 - y_2 = 2(2 - 4)$.

Simplifying the equation, we get: $y_1 - y_2 = 2(-2)$.

Evaluating the expression, we get: $y_1 - y_2 = -4$.

Therefore, the difference in y-values is $-4$.

Conclusion

Introduction

In our previous article, we discussed how to solve for the difference in y-values, denoted as $y_1 - y_2$, given the slope and the coordinates of two points on the line. In this article, we will provide a Q&A guide to help you better understand the concept and to address any questions you may have.

Q: What is the formula for slope?

A: The formula for slope is: $m = \frac{y_1 - y_2}{x_1 - x_2}$.

Q: How do I solve for the difference in y-values?

A: To solve for the difference in y-values, you can start by multiplying both sides of the equation by $x_1 - x_2$. This will eliminate the fraction and give you: $m(x_1 - x_2) = y_1 - y_2$.

Q: What if I have two points on a line, but I don't know the slope?

A: If you have two points on a line, but you don't know the slope, you can use the formula for slope to find it. Once you have the slope, you can use it to solve for the difference in y-values.

Q: Can I use the formula for slope to find the difference in y-values if I only know the coordinates of one point?

A: No, you cannot use the formula for slope to find the difference in y-values if you only know the coordinates of one point. You need to know the coordinates of two points on the line in order to use the formula for slope.

Q: What if the slope is zero?

A: If the slope is zero, then the line is horizontal, and the difference in y-values is equal to the difference in x-values.

Q: Can I use the formula for slope to find the difference in y-values if the slope is undefined?

A: No, you cannot use the formula for slope to find the difference in y-values if the slope is undefined. The slope is undefined when the line is vertical, and you cannot use the formula for slope to find the difference in y-values in this case.

Q: How do I know if the line is horizontal or vertical?

A: To determine if the line is horizontal or vertical, you can look at the slope. If the slope is zero, then the line is horizontal. If the slope is undefined, then the line is vertical.

Q: Can I use the formula for slope to find the difference in y-values if the line is a circle?

A: No, you cannot use the formula for slope to find the difference in y-values if the line is a circle. The formula for slope only works for lines, not for circles.

Q: What if I have a system of equations with two variables?

A: If you have a system of equations with two variables, you can use the formula for slope to find the difference in y-values. However, you will need to solve the system of equations first to find the values of the variables.

Conclusion

In this article, we have provided a Q&A guide to help you better understand how to solve for the difference in y-values, denoted as $y_1 - y_2$, given the slope and the coordinates of two points on the line. We have also addressed some common questions and provided examples to illustrate the concept.

Example Problems

Here are some example problems to help you practice solving for the difference in y-values:

  1. Find the difference in y-values for the points $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (4, 5)$, given that the slope is $m = 2$.
  2. Find the difference in y-values for the points $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (3, 4)$, given that the slope is $m = 1$.
  3. Find the difference in y-values for the points $(x_1, y_1) = (0, 0)$ and $(x_2, y_2) = (2, 3)$, given that the slope is $m = 0$.

Answers

Here are the answers to the example problems:

  1. y_1 - y_2 = 2(2 - 4) = 2(-2) = -4$.

  2. y_1 - y_2 = 1(1 - 3) = 1(-2) = -2$.

  3. y_1 - y_2 = 0(0 - 2) = 0$.

Conclusion

In this article, we have provided a Q&A guide to help you better understand how to solve for the difference in y-values, denoted as $y_1 - y_2$, given the slope and the coordinates of two points on the line. We have also addressed some common questions and provided examples to illustrate the concept.