Convert The Equations In The System Below Into Slope-intercept Form And Then Classify:${ \begin{array}{l} -3x - 8y = 11 \ -9x - 24y = 16 \end{array} }$In Slope-intercept Form, The First Equation Is { Y = \square $}$In

Introduction

In mathematics, the slope-intercept form of a linear equation is a fundamental concept used to represent lines on a coordinate plane. It is essential to convert equations into slope-intercept form to analyze and classify them. In this article, we will focus on converting the given system of equations into slope-intercept form and then classify them.

Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is represented as:

$$y = mx + b$$

where:

• $m$ is the slope of the line
• $b$ is the y-intercept

The slope-intercept form is a convenient way to represent lines, as it allows us to easily identify the slope and y-intercept of the line.

Converting the First Equation

The first equation in the given system is:

$$-3x - 8y = 11$$

To convert this equation into slope-intercept form, we need to isolate the variable $y$. We can do this by adding $8x$ to both sides of the equation and then dividing both sides by $-8$.

$$-8y = 11 + 3x$$

$$y = -\frac{3}{8}x - \frac{11}{8}$$

Therefore, the first equation in slope-intercept form is:

$$y = -\frac{3}{8}x - \frac{11}{8}$$

Converting the Second Equation

The second equation in the given system is:

$$-9x - 24y = 16$$

To convert this equation into slope-intercept form, we need to isolate the variable $y$. We can do this by adding $9x$ to both sides of the equation and then dividing both sides by $-24$.

$$-24y = 16 + 9x$$

$$y = -\frac{9}{24}x - \frac{16}{24}$$

Simplifying the equation, we get:

$$y = -\frac{3}{8}x - \frac{2}{3}$$

Therefore, the second equation in slope-intercept form is:

$$y = -\frac{3}{8}x - \frac{2}{3}$$

Classification of the Equations

Now that we have converted both equations into slope-intercept form, we can classify them. The two equations are:

$$y = -\frac{3}{8}x - \frac{11}{8}$$

$$y = -\frac{3}{8}x - \frac{2}{3}$$

We can see that both equations have the same slope, which is $-\frac{3}{8}$. However, they have different y-intercepts.

Parallel Lines

Since the two equations have the same slope but different y-intercepts, they represent parallel lines. Parallel lines are lines that lie in the same plane and never intersect.

Conclusion

In this article, we converted the given system of equations into slope-intercept form and then classified them. We found that the two equations represent parallel lines, as they have the same slope but different y-intercepts. Understanding the slope-intercept form of linear equations is essential in mathematics, and it allows us to analyze and classify lines on a coordinate plane.

Key Takeaways

• The slope-intercept form of a linear equation is represented as $y = mx + b$.
• To convert an equation into slope-intercept form, we need to isolate the variable $y$.
• Parallel lines have the same slope but different y-intercepts.

Further Reading

For further reading on linear equations and slope-intercept form, we recommend the following resources:

• Khan Academy: Linear Equations and Slope-Intercept Form
• Mathway: Linear Equations and Slope-Intercept Form
• Wolfram MathWorld: Linear Equations and Slope-Intercept Form
  Frequently Asked Questions (FAQs) on Converting Equations to Slope-Intercept Form and Classification =============================================================================================

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

A: The slope-intercept form of a linear equation is represented as:

$$y = mx + b$$

where:

• $m$ is the slope of the line
• $b$ is the y-intercept

Q: How do I convert an equation into slope-intercept form?

A: To convert an equation into slope-intercept form, you need to isolate the variable $y$. You can do this by adding or subtracting terms from both sides of the equation and then dividing both sides by the coefficient of $y$.

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

A: The slope represents the rate of change of the line, while the y-intercept represents the point where the line intersects the y-axis.

Q: How do I determine if two lines are parallel?

A: Two lines are parallel if they have the same slope but different y-intercepts.

Q: Can two lines have the same slope but different y-intercepts?

A: Yes, two lines can have the same slope but different y-intercepts. This means that they are parallel lines.

Q: What is the significance of the slope-intercept form in mathematics?

A: The slope-intercept form is a fundamental concept in mathematics, as it allows us to easily identify the slope and y-intercept of a line. This is essential in analyzing and classifying lines on a coordinate plane.

Q: How do I use the slope-intercept form to classify lines?

A: To classify lines using the slope-intercept form, you need to compare the slopes and y-intercepts of the lines. If two lines have the same slope but different y-intercepts, they are parallel lines.

Q: Can two lines have the same y-intercept but different slopes?

A: No, two lines cannot have the same y-intercept but different slopes. This would mean that they are the same line.

Q: What is the relationship between the slope and the y-intercept in the slope-intercept form?

A: The slope and the y-intercept are related in the slope-intercept form, as the slope represents the rate of change of the line, while the y-intercept represents the point where the line intersects the y-axis.

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

A: To find the equation of a line using the slope-intercept form, you need to know the slope and the y-intercept of the line. You can then use these values to write the equation of the line in slope-intercept form.

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

A: No, you cannot use the slope-intercept form to find the equation of a line if you only know the slope. You need to know the y-intercept as well.

Q: What is the significance of the y-intercept in the slope-intercept form?

A: The y-intercept represents the point where the line intersects the y-axis. This is an essential concept in mathematics, as it allows us to easily identify the y-intercept of a line.

Q: Can I use the slope-intercept form to find the equation of a line if I only know the y-intercept?

A: No, you cannot use the slope-intercept form to find the equation of a line if you only know the y-intercept. You need to know the slope as well.

Conclusion

In this article, we have answered some frequently asked questions on converting equations to slope-intercept form and classification. We have covered topics such as the slope-intercept form, converting equations, parallel lines, and the significance of the slope-intercept form in mathematics. We hope that this article has provided you with a better understanding of these concepts and has helped you to improve your knowledge of linear equations and slope-intercept form.