Find The $x$ And $y$ Intercepts Of The Following Linear Equation:$6x - 9y = 54$

by ADMIN 80 views

Introduction

In mathematics, the intercepts of a linear equation are the points at which the line intersects the x-axis and y-axis. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis. In this article, we will discuss how to find the x and y intercepts of a linear equation in the form ax+by=cax + by = c.

What are xx and yy Intercepts?

The x-intercept of a linear equation is the point at which the line crosses the x-axis. This occurs when the value of yy is equal to zero. The y-intercept of a linear equation is the point at which the line crosses the y-axis. This occurs when the value of xx is equal to zero.

Finding the xx Intercept

To find the x-intercept of a linear equation, we need to set the value of yy equal to zero and solve for xx. This is because the x-intercept occurs when the line crosses the x-axis, and the value of yy is equal to zero at this point.

Let's consider the linear equation 6x−9y=546x - 9y = 54. To find the x-intercept, we need to set the value of yy equal to zero and solve for xx.

# Import necessary modules
import sympy as sp

# Define variables
x = sp.symbols('x')
y = 0

# Define the linear equation
equation = 6*x - 9*y - 54

# Solve for x
solution = sp.solve(equation, x)

# Print the solution
print("The x-intercept is:", solution[0])

When we run this code, we get the following output:

The x-intercept is: 9

This means that the x-intercept of the linear equation 6x−9y=546x - 9y = 54 is the point (9, 0).

Finding the yy Intercept

To find the y-intercept of a linear equation, we need to set the value of xx equal to zero and solve for yy. This is because the y-intercept occurs when the line crosses the y-axis, and the value of xx is equal to zero at this point.

Let's consider the linear equation 6x−9y=546x - 9y = 54. To find the y-intercept, we need to set the value of xx equal to zero and solve for yy.

# Import necessary modules
import sympy as sp

# Define variables
x = 0
y = sp.symbols('y')

# Define the linear equation
equation = 6*x - 9*y - 54

# Solve for y
solution = sp.solve(equation, y)

# Print the solution
print("The y-intercept is:", solution[0])

When we run this code, we get the following output:

The y-intercept is: -6

This means that the y-intercept of the linear equation 6x−9y=546x - 9y = 54 is the point (0, -6).

Conclusion

In this article, we discussed how to find the x and y intercepts of a linear equation in the form ax+by=cax + by = c. We used the example of the linear equation 6x−9y=546x - 9y = 54 to illustrate the process of finding the x and y intercepts. We showed how to use Python code to solve for the x and y intercepts, and we provided the solutions to the example equation.

Example Use Cases

Finding the x and y intercepts of a linear equation has many practical applications in mathematics and science. Here are a few example use cases:

  • Graphing linear equations: Finding the x and y intercepts of a linear equation is essential for graphing the equation on a coordinate plane.
  • Solving systems of linear equations: Finding the x and y intercepts of a linear equation is a crucial step in solving systems of linear equations.
  • Modeling real-world phenomena: Linear equations are often used to model real-world phenomena, such as the motion of objects or the growth of populations. Finding the x and y intercepts of a linear equation can help us understand these phenomena.

Tips and Tricks

Here are a few tips and tricks for finding the x and y intercepts of a linear equation:

  • Use the equation in the form ax+by=cax + by = c: This form makes it easy to identify the coefficients of xx and yy and to solve for the intercepts.
  • Set the value of yy equal to zero to find the x-intercept: This is because the x-intercept occurs when the line crosses the x-axis, and the value of yy is equal to zero at this point.
  • Set the value of xx equal to zero to find the y-intercept: This is because the y-intercept occurs when the line crosses the y-axis, and the value of xx is equal to zero at this point.

Q: What is the difference between the xx and yy intercepts of a linear equation?

A: The x-intercept of a linear equation is the point at which the line crosses the x-axis, and the value of yy is equal to zero. The y-intercept of a linear equation is the point at which the line crosses the y-axis, and the value of xx is equal to zero.

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

A: To find the x-intercept of a linear equation, you need to set the value of yy equal to zero and solve for xx. This is because the x-intercept occurs when the line crosses the x-axis, and the value of yy is equal to zero at this point.

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

A: To find the y-intercept of a linear equation, you need to set the value of xx equal to zero and solve for yy. This is because the y-intercept occurs when the line crosses the y-axis, and the value of xx is equal to zero at this point.

Q: What is the formula for finding the xx and yy intercepts of a linear equation?

A: The formula for finding the x-intercept of a linear equation is:

x=cax = \frac{c}{a}

where aa is the coefficient of xx, cc is the constant term, and xx is the x-intercept.

The formula for finding the y-intercept of a linear equation is:

y=cby = \frac{c}{b}

where bb is the coefficient of yy, cc is the constant term, and yy is the y-intercept.

Q: Can I use a calculator to find the xx and yy intercepts of a linear equation?

A: Yes, you can use a calculator to find the x and y intercepts of a linear equation. Most graphing calculators have a built-in function for finding the x and y intercepts of a linear equation.

Q: How do I graph a linear equation on a coordinate plane?

A: To graph a linear equation on a coordinate plane, you need to find the x and y intercepts of the equation and plot the points on the coordinate plane. You can then draw a line through the points to represent the linear equation.

Q: What are some real-world applications of finding the xx and yy intercepts of a linear equation?

A: Finding the x and y intercepts of a linear equation has many real-world applications, including:

  • Graphing linear equations: Finding the x and y intercepts of a linear equation is essential for graphing the equation on a coordinate plane.
  • Solving systems of linear equations: Finding the x and y intercepts of a linear equation is a crucial step in solving systems of linear equations.
  • Modeling real-world phenomena: Linear equations are often used to model real-world phenomena, such as the motion of objects or the growth of populations. Finding the x and y intercepts of a linear equation can help us understand these phenomena.

Q: What are some common mistakes to avoid when finding the xx and yy intercepts of a linear equation?

A: Some common mistakes to avoid when finding the x and y intercepts of a linear equation include:

  • Not setting the value of yy equal to zero to find the x-intercept: This is because the x-intercept occurs when the line crosses the x-axis, and the value of yy is equal to zero at this point.
  • Not setting the value of xx equal to zero to find the y-intercept: This is because the y-intercept occurs when the line crosses the y-axis, and the value of xx is equal to zero at this point.
  • Not using the correct formula for finding the x and y intercepts: Make sure to use the correct formula for finding the x and y intercepts, which is x=cax = \frac{c}{a} and y=cby = \frac{c}{b}.

By following these tips and avoiding common mistakes, you can easily find the x and y intercepts of a linear equation and apply this knowledge to a wide range of mathematical and scientific problems.