$\frac{1}{3}x - 4y = 12$a. Show How The $x$-intercept Is $(36, 0$\].b. Show How The $y$-intercept Is $(0, -3$\].

Introduction

In mathematics, the $x$-intercept and $y$-intercept are two important concepts related to linear equations. The $x$-intercept is the point where the graph of a linear equation intersects the $x$-axis, while the $y$-intercept is the point where the graph intersects the $y$-axis. In this article, we will discuss how to find the $x$-intercept and $y$-intercept of a linear equation in the form $\frac{1}{3}x - 4y = 12$.

What is the $x$-Intercept?

The $x$-intercept is the point where the graph of a linear equation intersects the $x$-axis. In other words, it is the point where the value of $y$ is equal to zero. To find the $x$-intercept of a linear equation, we need to set the value of $y$ to zero and solve for $x$.

Finding the $x$-Intercept of the Given Equation

The given equation is $\frac{1}{3}x - 4y = 12$. To find the $x$-intercept, we need to set the value of $y$ to zero and solve for $x$. We can do this by substituting $y = 0$ into the equation and solving for $x$.

from sympy import symbols, Eq, solve

# Define the variables
x, y = symbols('x y')

# Define the equation
equation = Eq((1/3)*x - 4*y, 12)

# Substitute y = 0 into the equation
equation_x_intercept = equation.subs(y, 0)

# Solve for x
solution = solve(equation_x_intercept, x)

print(solution)

When we run this code, we get the solution $x = 36$. This means that the $x$-intercept of the given equation is $(36, 0)$.

What is the $y$-Intercept?

The $y$-intercept is the point where the graph of a linear equation intersects the $y$-axis. In other words, it is the point where the value of $x$ is equal to zero. To find the $y$-intercept of a linear equation, we need to set the value of $x$ to zero and solve for $y$.

Finding the $y$-Intercept of the Given Equation

The given equation is $\frac{1}{3}x - 4y = 12$. To find the $y$-intercept, we need to set the value of $x$ to zero and solve for $y$. We can do this by substituting $x = 0$ into the equation and solving for $y$.

from sympy import symbols, Eq, solve

# Define the variables
x, y = symbols('x y')

# Define the equation
equation = Eq((1/3)*x - 4*y, 12)

# Substitute x = 0 into the equation
equation_y_intercept = equation.subs(x, 0)

# Solve for y
solution = solve(equation_y_intercept, y)

print(solution)

When we run this code, we get the solution $y = -3$. This means that the $y$-intercept of the given equation is $(0, -3)$.

Conclusion

In this article, we discussed how to find the $x$-intercept and $y$-intercept of a linear equation in the form $\frac{1}{3}x - 4y = 12$. We used the concept of setting the value of $y$ to zero to find the $x$-intercept and setting the value of $x$ to zero to find the $y$-intercept. We also used Python code to solve for the $x$-intercept and $y$-intercept. The $x$-intercept of the given equation is $(36, 0)$, and the $y$-intercept is $(0, -3)$.

References

• [1] Khan Academy. (n.d.). Linear Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f7d
• [2] Math Open Reference. (n.d.). Linear Equations. Retrieved from https://www.mathopenref.com/linearequations.html

Frequently Asked Questions

• Q: What is the $x$-intercept of a linear equation? A: The $x$-intercept is the point where the graph of a linear equation intersects the $x$-axis.
• Q: What is the $y$-intercept of a linear equation? A: The $y$-intercept is the point where the graph of a linear equation intersects the $y$-axis.
• Q: How do I find the $x$-intercept of a linear equation? A: To find the $x$-intercept, set the value of $y$ to zero and solve for $x$.
• Q: How do I find the $y$-intercept of a linear equation? A: To find the $y$-intercept, set the value of $x$ to zero and solve for $y$.
  Frequently Asked Questions (FAQs) about Linear Equations ===========================================================

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.

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

A: The $x$-intercept is the point where the graph of a linear equation intersects the $x$-axis. It is the value of $x$ when $y$ is equal to zero.

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

A: To find the $x$-intercept, set the value of $y$ to zero and solve for $x$. This can be done by substituting $y = 0$ into the equation and solving for $x$.

Q: What is the $y$-intercept of a linear equation?

A: The $y$-intercept is the point where the graph of a linear equation 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 linear equation?

A: To find the $y$-intercept, set the value of $x$ to zero and solve for $y$. This can be done by substituting $x = 0$ into the equation and solving for $y$.

Q: What is the slope of a linear equation?

A: The slope of a linear equation is a measure of how steep the line is. It is calculated by dividing the change in $y$ by the change in $x$.

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

A: To find the slope, use 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.

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

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

Q: How do I graph a linear equation?

A: To graph a linear equation, first find the $x$-intercept and $y$-intercept. Then, plot these points on a coordinate plane and draw a line through them.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, use either the substitution method or the elimination method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: What is the importance of linear equations in real-life situations?

A: Linear equations are used in a wide range of real-life situations, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the flow of fluids, and the growth of populations.

Q: How do I use linear equations in my daily life?

A: Linear equations are used in many everyday situations, such as calculating the cost of goods, determining the amount of time it takes to complete a task, and predicting the outcome of a game or competition.

Conclusion

In this article, we have answered some of the most frequently asked questions about linear equations. We have covered topics such as the $x$-intercept and $y$-intercept, the slope, and the equation of a line in slope-intercept form. We have also discussed how to graph a linear equation and how to solve a system of linear equations. We hope that this article has been helpful in answering your questions about linear equations.