Find The Slope And The Y Y Y -intercept Of The Line With The Given Equation. X − 2 Y = − 8 X - 2y = -8 X − 2 Y = − 8 Slope: □ \square □ Y Y Y -intercept: ( X , Y ) = □ (x, Y) = \square ( X , Y ) = □

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Understanding the Problem

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In this article, we will delve into the world of linear equations and explore how to find the slope and $y$-intercept of a given line. The equation $x - 2y = -8$ is a linear equation in two variables, $x$ and $y$. Our goal is to find the slope and $y$-intercept of this line.

What is the Slope?

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The slope of a line is a measure of how steep it is. 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 often denoted by the letter $m$ and can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

What is the $y$-Intercept?

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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 zero. The $y$-intercept is often denoted by the letter $b$ and can be calculated using the formula:

$$b = y_1$$

where $(x_1, y_1)$ is a point on the line.

Finding the Slope and $y$-Intercept of the Given Equation

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To find the slope and $y$-intercept of the given equation $x - 2y = -8$, we need to rewrite the equation in the slope-intercept form, which is:

$$y = mx + b$$

where $m$ is the slope and $b$ is the $y$-intercept.

Rewriting the Equation in Slope-Intercept Form

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To rewrite the equation in slope-intercept form, we need to isolate $y$ on one side of the equation. We can do this by adding $2y$ to both sides of the equation and then dividing both sides by $2$.

$$x - 2y = -8$$

$$2y = x + 8$$

$$y = \frac{x + 8}{2}$$

Finding the Slope

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Now that we have rewritten the equation in slope-intercept form, we can see that the slope is the coefficient of $x$, which is $\frac{1}{2}$.

Finding the $y$-Intercept

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To find the $y$-intercept, we need to find the value of $y$ when $x$ is equal to zero. We can do this by substituting $x = 0$ into the equation.

$$y = \frac{0 + 8}{2}$$

$$y = 4$$

Conclusion

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In this article, we have learned how to find the slope and $y$-intercept of a linear equation. We have rewritten the equation $x - 2y = -8$ in slope-intercept form and found the slope to be $\frac{1}{2}$ and the $y$-intercept to be $(0, 4)$.

Example Problems

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Problem 1

Find the slope and $y$-intercept of the line with the equation $2x - 3y = 6$.

Solution

To find the slope and $y$-intercept, we need to rewrite the equation in slope-intercept form.

$$2x - 3y = 6$$

$$3y = 2x - 6$$

$$y = \frac{2x - 6}{3}$$

The slope is the coefficient of $x$, which is $\frac{2}{3}$. The $y$-intercept is the value of $y$ when $x$ is equal to zero, which is $-2$.

Problem 2

Find the slope and $y$-intercept of the line with the equation $x + 4y = 10$.

Solution

To find the slope and $y$-intercept, we need to rewrite the equation in slope-intercept form.

$$x + 4y = 10$$

$$4y = -x + 10$$

$$y = \frac{-x + 10}{4}$$

The slope is the coefficient of $x$, which is $-\frac{1}{4}$. The $y$-intercept is the value of $y$ when $x$ is equal to zero, which is $\frac{5}{2}$.

Tips and Tricks

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• To find the slope and $y$-intercept of a linear equation, you need to rewrite the equation in slope-intercept form.
• The slope is the coefficient of $x$ in the slope-intercept form of the equation.
• The $y$-intercept is the value of $y$ when $x$ is equal to zero in the slope-intercept form of the equation.

Conclusion

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In this article, we have learned how to find the slope and $y$-intercept of a linear equation. We have rewritten the equation $x - 2y = -8$ in slope-intercept form and found the slope to be $\frac{1}{2}$ and the $y$-intercept to be $(0, 4)$. We have also provided example problems and tips and tricks to help you understand the concept better.

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Q1: What is the slope-intercept form of a linear equation?

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A1: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Q2: How do I find the slope of a linear equation?

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A2: To find the slope of a linear equation, you need to rewrite the equation in slope-intercept form. The slope is the coefficient of $x$ in the slope-intercept form of the equation.

Q3: How do I find the $y$-intercept of a linear equation?

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A3: To find the $y$-intercept of a linear equation, you need to rewrite the equation in slope-intercept form. The $y$-intercept is the value of $y$ when $x$ is equal to zero in the slope-intercept form of the equation.

Q4: What is the difference between the slope and the $y$-intercept?

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A4: The slope is a measure of how steep a line is, while the $y$-intercept is the point where the line intersects the $y$-axis.

Q5: How do I rewrite a linear equation in slope-intercept form?

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A5: To rewrite a linear equation in slope-intercept form, you need to isolate $y$ on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, and then dividing both sides by the coefficient of $y$.

Q6: What is the significance of the slope and $y$-intercept in real-life applications?

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A6: The slope and $y$-intercept are important in real-life applications such as physics, engineering, and economics. They are used to model and analyze the behavior of objects and systems.

Q7: Can I find the slope and $y$-intercept of a non-linear equation?

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A7: No, you cannot find the slope and $y$-intercept of a non-linear equation. The slope and $y$-intercept are properties of linear equations, and they do not apply to non-linear equations.

Q8: How do I graph a linear equation using its slope and $y$-intercept?

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A8: To graph a linear equation using its slope and $y$-intercept, you need to plot the $y$-intercept on the $y$-axis, and then use the slope to determine the direction and steepness of the line.

Q9: Can I find the slope and $y$-intercept of a linear equation with a fractional coefficient?

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A9: Yes, you can find the slope and $y$-intercept of a linear equation with a fractional coefficient. You can rewrite the equation in slope-intercept form, and then find the slope and $y$-intercept as usual.

Q10: How do I check my work when finding the slope and $y$-intercept of a linear equation?

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A10: To check your work when finding the slope and $y$-intercept of a linear equation, you can substitute the slope and $y$-intercept back into the original equation to see if it is true. You can also graph the equation using the slope and $y$-intercept to see if it matches the original equation.

Conclusion

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In this article, we have answered frequently asked questions about finding the slope and $y$-intercept of a linear equation. We have provided explanations and examples to help you understand the concept better.