Find The Slope Of The Line Determined By The Equation:${ 8x - Y = 24 }$

Mar 11, 2025 by ADMIN 73 views

**Finding the Slope of a Line from an Equation**

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Introduction

In mathematics, the slope of a line is a fundamental concept used to describe the steepness or incline of a line. It is a crucial element in graphing and analyzing linear equations. In this article, we will explore how to find the slope of a line determined by a given equation. We will use the equation $8x - y = 24$ as an example to demonstrate the process.

Understanding the Equation

The given equation is $8x - y = 24$ . To find the slope of this line, 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

To rewrite the equation in slope-intercept form, we need to isolate the variable $y$ on one side of the equation. We can do this by subtracting $8x$ from both sides of the equation and then multiplying both sides by $-1$ .

${ 8x - y = 24 }$

Subtracting $8x$ from both sides:

${ -y = -8x + 24 }$

Multiplying both sides by $-1$ :

${ y = 8x - 24 }$

Identifying the Slope

Now that we have rewritten the equation in slope-intercept form, we can identify the slope. In the equation $y = 8x - 24$ , the slope is the coefficient of the $x$ term, which is $8$ . Therefore, the slope of the line determined by the equation $8x - y = 24$ is $8$ .

Interpreting the Slope

The slope of a line can be interpreted as the rate of change of the dependent variable (in this case, $y$ ) with respect to the independent variable (in this case, $x$ ). In other words, it represents how much $y$ changes when $x$ changes by one unit. In this case, the slope of $8$ means that for every one-unit increase in $x$ , $y$ increases by $8$ units.

Real-World Applications

The concept of slope has numerous real-world applications. For example, in physics, the slope of a line can represent the rate of change of velocity or acceleration. In economics, the slope of a line can represent the rate of change of demand or supply. In engineering, the slope of a line can represent the rate of change of a physical quantity, such as temperature or pressure.

Conclusion

In conclusion, finding the slope of a line from an equation is a straightforward process that involves rewriting the equation in slope-intercept form and identifying the coefficient of the $x$ term. The slope of a line can be interpreted as the rate of change of the dependent variable with respect to the independent variable. The concept of slope has numerous real-world applications and is a fundamental element in graphing and analyzing linear equations.

Example Problems

Problem 1

Find the slope of the line determined by the equation $3x + 2y = 12$ .

Solution

To find the slope of the line, we need to rewrite the equation in slope-intercept form. We can do this by subtracting $3x$ from both sides of the equation and then multiplying both sides by $-1/2$ .

${ 3x + 2y = 12 }$

Subtracting $3x$ from both sides:

${ 2y = -3x + 12 }$

Multiplying both sides by $-1/2$ :

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

The slope of the line is the coefficient of the $x$ term, which is $-\frac{3}{2}$ .

Problem 2

Find the slope of the line determined by the equation $x - 4y = 8$ .

Solution

To find the slope of the line, we need to rewrite the equation in slope-intercept form. We can do this by subtracting $x$ from both sides of the equation and then multiplying both sides by $-1/4$ .

${ x - 4y = 8 }$

Subtracting $x$ from both sides:

${ -4y = -x + 8 }$

Multiplying both sides by $-1/4$ :

${ y = \frac{1}{4}x - 2 }$

The slope of the line is the coefficient of the $x$ term, which is $\frac{1}{4}$ .

Tips and Tricks

To find the slope of a line from an equation, rewrite the equation in slope-intercept form.
The slope of a line is the coefficient of the $x$ term in the slope-intercept form of the equation.
The slope of a line can be interpreted as the rate of change of the dependent variable with respect to the independent variable.
The concept of slope has numerous real-world applications.

Conclusion

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Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It is a ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

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

A: To find the slope of a line from an equation, you need to rewrite the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. The slope is the coefficient of the x term.

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

A: The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I rewrite an equation in slope-intercept form?

A: To rewrite an equation in slope-intercept form, you need to isolate the variable y on one side of the equation. You can do this by subtracting the x term from both sides of the equation and then multiplying both sides by the reciprocal of the coefficient of the x term.

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 zero.

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

A: To find the y-intercept of a line, you need to set x equal to zero in the equation and solve for y.

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

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

Q: Can a line have a slope of zero?

A: Yes, a line can have a slope of zero. This means that the line is horizontal and does not change in the vertical direction.

Q: Can a line have a slope of infinity?

A: No, a line cannot have a slope of infinity. However, a line can have a very steep slope, which can be represented by a large positive or negative number.

Q: How do I graph a line with a given slope and y-intercept?

A: To graph a line with a given slope and y-intercept, you need to use the slope-intercept form of the equation (y = mx + b) and plot the point (0, b) on the y-axis. Then, use the slope to find another point on the line and plot it. Finally, draw a line through the two points.

Q: What is the significance of the slope of a line in real-world applications?

A: The slope of a line is significant in real-world applications because it represents the rate of change of a quantity. For example, in physics, the slope of a line can represent the rate of change of velocity or acceleration. In economics, the slope of a line can represent the rate of change of demand or supply.

Q: Can I use the slope of a line to predict future values?

A: Yes, you can use the slope of a line to predict future values. If you know the slope of a line and the current value of the quantity, you can use the slope to predict the future value of the quantity.

Q: How do I use the slope of a line to make predictions?

A: To use the slope of a line to make predictions, you need to use the slope-intercept form of the equation (y = mx + b) and plug in the current value of x and the slope m. Then, solve for y to get the predicted value.

Q: What are some common mistakes to avoid when finding the slope of a line?

A: Some common mistakes to avoid when finding the slope of a line include:

Not rewriting the equation in slope-intercept form
Not isolating the variable y on one side of the equation
Not using the correct coefficient of the x term as the slope
Not plotting the correct points on the graph
Not using the correct slope to make predictions

Q: How do I check my work when finding the slope of a line?

A: To check your work when finding the slope of a line, you need to:

Rewrite the equation in slope-intercept form
Identify the coefficient of the x term as the slope
Plot the correct points on the graph
Use the slope to make predictions
Check your answers against the original equation

Q: What are some real-world applications of the slope of a line?

A: Some real-world applications of the slope of a line include:

Physics: The slope of a line can represent the rate of change of velocity or acceleration.
Economics: The slope of a line can represent the rate of change of demand or supply.
Engineering: The slope of a line can represent the rate of change of a physical quantity, such as temperature or pressure.
Finance: The slope of a line can represent the rate of change of a financial quantity, such as stock prices or interest rates.

Q: Can I use the slope of a line to compare different quantities?

A: Yes, you can use the slope of a line to compare different quantities. If you know the slope of a line and the current value of the quantity, you can use the slope to compare the quantity to other quantities.

Q: How do I use the slope of a line to compare different quantities?

A: To use the slope of a line to compare different quantities, you need to use the slope-intercept form of the equation (y = mx + b) and plug in the current value of x and the slope m. Then, solve for y to get the predicted value. You can then compare the predicted value to other quantities.

Q: What are some common misconceptions about the slope of a line?

A: Some common misconceptions about the slope of a line include:

Thinking that the slope of a line is always positive
Thinking that the slope of a line is always negative
Thinking that the slope of a line is always zero
Thinking that the slope of a line is always infinity

Q: How do I avoid common misconceptions about the slope of a line?

A: To avoid common misconceptions about the slope of a line, you need to:

Understand the definition of the slope of a line
Understand how to find the slope of a line
Understand how to use the slope of a line to make predictions
Understand how to compare different quantities using the slope of a line

Q: What are some advanced topics related to the slope of a line?

A: Some advanced topics related to the slope of a line include:

Calculus: The slope of a line can be used to find the derivative of a function.
Differential equations: The slope of a line can be used to solve differential equations.
Linear algebra: The slope of a line can be used to find the inverse of a matrix.
Statistics: The slope of a line can be used to find the correlation coefficient between two variables.

Q: How do I apply the slope of a line to advanced topics?

A: To apply the slope of a line to advanced topics, you need to:

Understand the definition of the slope of a line
Understand how to find the slope of a line
Understand how to use the slope of a line to make predictions
Understand how to compare different quantities using the slope of a line
Apply the slope of a line to advanced topics, such as calculus, differential equations, linear algebra, and statistics.