For The Linear Function $1.5x - 3y = 39$, Find The Intercepts.a. Show How The $x$-intercept Is \[$(26, 0)\$\].b. Show How The $y$-intercept Is \[$(0, -13)\$\].

Introduction

In mathematics, a linear function is a polynomial function of degree one, which means it has the form $f(x) = ax + b$, where $a$ and $b$ are constants. The linear function $1.5x - 3y = 39$ is a specific example of a linear function in two variables, $x$ and $y$. In this article, we will explore how to find the intercepts of this linear function.

What are Intercepts?

Intercepts are the points where the graph of a linear function intersects the $x$-axis and the $y$-axis. The $x$-intercept is the point where the graph intersects the $x$-axis, and the $y$-intercept is the point where the graph intersects the $y$-axis.

Finding the $x$-Intercept

To find the $x$-intercept of the linear function $1.5x - 3y = 39$, we need to set $y$ equal to zero and solve for $x$. This is because the $x$-intercept occurs when the graph intersects the $x$-axis, and the $y$-coordinate is always zero at this point.

# Define the equation
def equation(x, y):
    return 1.5*x - 3*y - 39

# Set y equal to zero and solve for x
x_intercept = 39 / 1.5
print(f"The x-intercept is ({x_intercept}, 0)")

Solution

The $x$-intercept is found by setting $y$ equal to zero and solving for $x$. In this case, we have:

$$1.5x - 3(0) = 39$$

Simplifying the equation, we get:

$$1.5x = 39$$

Dividing both sides by 1.5, we get:

$$x = 26$$

Therefore, the $x$-intercept is $(26, 0)$.

Finding the $y$-Intercept

To find the $y$-intercept of the linear function $1.5x - 3y = 39$, we need to set $x$ equal to zero and solve for $y$. This is because the $y$-intercept occurs when the graph intersects the $y$-axis, and the $x$-coordinate is always zero at this point.

# Define the equation
def equation(x, y):
    return 1.5*x - 3*y - 39

# Set x equal to zero and solve for y
y_intercept = -39 / 3
print(f"The y-intercept is (0, {y_intercept})")

Solution

The $y$-intercept is found by setting $x$ equal to zero and solving for $y$. In this case, we have:

$$1.5(0) - 3y = 39$$

Simplifying the equation, we get:

$$-3y = 39$$

Dividing both sides by -3, we get:

$$y = -13$$

Therefore, the $y$-intercept is $(0, -13)$.

Conclusion

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Q&A: Linear Function Intercepts

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

A: The $x$-intercept is the point where the graph of a linear function intersects the $x$-axis, and the $y$-intercept is the point where the graph intersects the $y$-axis.

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

A: To find the $x$-intercept, set $y$ equal to zero and solve for $x$. This is because the $x$-intercept occurs when the graph intersects the $x$-axis, and the $y$-coordinate is always zero at this point.

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

A: To find the $y$-intercept, set $x$ equal to zero and solve for $y$. This is because the $y$-intercept occurs when the graph intersects the $y$-axis, and the $x$-coordinate is always zero at this point.

Q: What is the formula for finding the $x$-intercept?

A: The formula for finding the $x$-intercept is:

$$x = \frac{b}{a}$$

where $a$ and $b$ are the coefficients of the linear function.

Q: What is the formula for finding the $y$-intercept?

A: The formula for finding the $y$-intercept is:

$$y = \frac{-b}{a}$$

where $a$ and $b$ are the coefficients of the linear function.

Q: Can I use a graphing calculator to find the intercepts of a linear function?

A: Yes, you can use a graphing calculator to find the intercepts of a linear function. Simply graph the function and use the calculator's built-in features to find the $x$-intercept and $y$-intercept.

Q: What if I have a linear function in the form $y = mx + b$? How do I find the intercepts?

A: To find the $x$-intercept, set $y$ equal to zero and solve for $x$. To find the $y$-intercept, set $x$ equal to zero and solve for $y$. The formulas for finding the intercepts are the same as before:

$$x = \frac{-b}{m}$$

$$y = \frac{-b}{m}$$

Q: Can I use the slope-intercept form of a linear function to find the intercepts?

A: Yes, you can use the slope-intercept form of a linear function to find the intercepts. The slope-intercept form is:

$$y = mx + b$$

where $m$ is the slope and $b$ is the $y$-intercept. To find the $x$-intercept, set $y$ equal to zero and solve for $x$. To find the $y$-intercept, simply read the value of $b$.

Conclusion

In this article, we have answered some common questions about linear function intercepts. We have covered topics such as finding the $x$-intercept and $y$-intercept, using a graphing calculator, and using the slope-intercept form of a linear function. We hope this article has been helpful in understanding linear function intercepts.