Find The $x$-intercept And $y$-intercept Of The Line.$\[ 5x - 2y = 10 \\]$x$-intercept: \[$\square\$\]$y$-intercept: \[$\square\$\]

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Introduction

In mathematics, the $x$-intercept and $y$-intercept of a line are two important concepts that help us understand the behavior of a line on a coordinate plane. The $x$-intercept is the point where the line intersects the $x$-axis, while the $y$-intercept is the point where the line intersects the $y$-axis. In this article, we will learn how to find the $x$-intercept and $y$-intercept of a line using a given equation.

What are $x$-intercept and $y$-intercept?

The $x$-intercept of a line is the point where the line intersects the $x$-axis. This means that at this point, the value of $y$ is zero. Similarly, the $y$-intercept of a line is the point where the line intersects the $y$-axis. This means that at this point, the value of $x$ is zero.

Finding the $x$-intercept

To find the $x$-intercept of a line, we need to set the value of $y$ to zero and solve for $x$. Let's consider the given equation:

${ 5x - 2y = 10 }$

We can set the value of $y$ to zero and solve for $x$ as follows:

${ 5x - 2(0) = 10 }$

${ 5x = 10 }$

${ x = \frac{10}{5} }$

${ x = 2 }$

Therefore, the $x$-intercept of the line is $(2, 0)$.

Finding the $y$-intercept

To find the $y$-intercept of a line, we need to set the value of $x$ to zero and solve for $y$. Let's consider the given equation:

${ 5x - 2y = 10 }$

We can set the value of $x$ to zero and solve for $y$ as follows:

${ 5(0) - 2y = 10 }$

${ -2y = 10 }$

${ y = -\frac{10}{2} }$

${ y = -5 }$

Therefore, the $y$-intercept of the line is $(0, -5)$.

Conclusion

In this article, we learned how to find the $x$-intercept and $y$-intercept of a line using a given equation. We set the value of $y$ to zero to find the $x$-intercept and the value of $x$ to zero to find the $y$-intercept. We used the given equation to solve for the values of $x$ and $y$. We found that the $x$-intercept of the line is $(2, 0)$ and the $y$-intercept of the line is $(0, -5)$.

Example Problems

1. Find the $x$-intercept and $y$-intercept of the line given by the equation $3x + 2y = 6$.
2. Find the $x$-intercept and $y$-intercept of the line given by the equation $2x - 3y = 9$.

Solutions

1. To find the $x$-intercept, we set the value of $y$ to zero and solve for $x$:

${ 3x + 2(0) = 6 }$

${ 3x = 6 }$

${ x = \frac{6}{3} }$

${ x = 2 }$

Therefore, the $x$-intercept of the line is $(2, 0)$.

To find the $y$-intercept, we set the value of $x$ to zero and solve for $y$:

${ 3(0) + 2y = 6 }$

${ 2y = 6 }$

${ y = \frac{6}{2} }$

${ y = 3 }$

Therefore, the $y$-intercept of the line is $(0, 3)$.

2. To find the $x$-intercept, we set the value of $y$ to zero and solve for $x$:

${ 2x - 3(0) = 9 }$

${ 2x = 9 }$

${ x = \frac{9}{2} }$

${ x = 4.5 }$

Therefore, the $x$-intercept of the line is $(4.5, 0)$.

To find the $y$-intercept, we set the value of $x$ to zero and solve for $y$:

${ 2(0) - 3y = 9 }$

${ -3y = 9 }$

${ y = -\frac{9}{3} }$

${ y = -3 }$

Therefore, the $y$-intercept of the line is $(0, -3)$.

Final Thoughts

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Frequently Asked Questions

Q: What is the $x$-intercept of a line?

A: The $x$-intercept of a line is the point where the line intersects the $x$-axis. This means that at this point, the value of $y$ is zero.

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. This means that at this point, the value of $x$ is zero.

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

A: To find the $x$-intercept of a line, you need to set the value of $y$ to zero and solve for $x$. You can use the given equation to solve for the value of $x$.

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

A: To find the $y$-intercept of a line, you need to set the value of $x$ to zero and solve for $y$. You can use the given equation to solve for the value of $y$.

Q: What if the given equation is not in the form $y = mx + b$?

A: If the given equation is not in the form $y = mx + b$, you can still find the $x$-intercept and $y$-intercept by setting the value of $y$ to zero and solving for $x$, and setting the value of $x$ to zero and solving for $y$.

Q: Can I use a graphing calculator to find the $x$-intercept and $y$-intercept of a line?

A: Yes, you can use a graphing calculator to find the $x$-intercept and $y$-intercept of a line. Simply graph the line and use the calculator to find the points where the line intersects the $x$-axis and $y$-axis.

Q: What if the line is not a straight line?

A: If the line is not a straight line, you may need to use a different method to find the $x$-intercept and $y$-intercept. For example, you may need to use a graphing calculator or a computer program to find the points where the line intersects the $x$-axis and $y$-axis.

Q: Can I find the $x$-intercept and $y$-intercept of a line using a table of values?

A: Yes, you can find the $x$-intercept and $y$-intercept of a line using a table of values. Simply create a table with the values of $x$ and $y$ and use it to find the points where the line intersects the $x$-axis and $y$-axis.

Q: What if I make a mistake when finding the $x$-intercept and $y$-intercept of a line?

A: If you make a mistake when finding the $x$-intercept and $y$-intercept of a line, you can try reworking the problem or using a different method to find the correct answers.

Common Mistakes

Mistake 1: Not setting the value of $y$ to zero to find the $x$-intercept

A: Make sure to set the value of $y$ to zero to find the $x$-intercept.

Mistake 2: Not setting the value of $x$ to zero to find the $y$-intercept

A: Make sure to set the value of $x$ to zero to find the $y$-intercept.

Mistake 3: Not using the given equation to solve for the value of $x$ or $y$

A: Make sure to use the given equation to solve for the value of $x$ or $y$.

Tips and Tricks

Tip 1: Use a graphing calculator to find the $x$-intercept and $y$-intercept of a line

A: Using a graphing calculator can make it easier to find the $x$-intercept and $y$-intercept of a line.

Tip 2: Create a table of values to find the $x$-intercept and $y$-intercept of a line

A: Creating a table of values can make it easier to find the $x$-intercept and $y$-intercept of a line.

Tip 3: Check your work to make sure you have the correct answers

A: Make sure to check your work to make sure you have the correct answers.

Conclusion

Finding the $x$-intercept and $y$-intercept of a line is an important concept in mathematics. By following the steps outlined in this article, you can find the $x$-intercept and $y$-intercept of a line using a given equation. Remember to set the value of $y$ to zero to find the $x$-intercept and the value of $x$ to zero to find the $y$-intercept. You can also use a graphing calculator or a table of values to find the $x$-intercept and $y$-intercept of a line.