Find The \[$ X \$\]- And \[$ Y \$\]-intercepts Of The Line. Write Ordered Pairs Representing The Points Where The Line Crosses The Axes.$\[ 4x + 6y - 15 = 0 \\]

Feb 28, 2025 by ADMIN 161 views

**Finding the x- and y-Intercepts of a Linear Equation**

Introduction

In mathematics, the intercepts of a linear equation are the points where the line crosses the x-axis and the y-axis. These points are represented by ordered pairs, where the first value is the x-coordinate and the second value is the y-coordinate. In this article, we will learn how to find the x- and y-intercepts of a linear equation in the form of ax + by + c = 0.

Understanding the Equation

The given equation is 4x + 6y - 15 = 0. To find the x- and y-intercepts, we need to understand the concept of intercepts. 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.

Finding the x-Intercept

To find the x-intercept, we need to set y = 0 and solve for x. This is because the x-intercept occurs when the line crosses the x-axis, and at this point, the y-coordinate is always 0.

# Import necessary modules
import sympy as sp

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

# Substitute y = 0 into the equation
equation = 4*x + 6*y - 15
equation = equation.subs(y, 0)

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

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

Finding the y-Intercept

To find the y-intercept, we need to set x = 0 and solve for y. This is because the y-intercept occurs when the line crosses the y-axis, and at this point, the x-coordinate is always 0.

# Import necessary modules
import sympy as sp

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

# Substitute x = 0 into the equation
equation = 4*x + 6*y - 15
equation = equation.subs(x, 0)

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

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

Conclusion

In this article, we learned how to find the x- and y-intercepts of a linear equation in the form of ax + by + c = 0. We used the concept of intercepts and substituted y = 0 to find the x-intercept and x = 0 to find the y-intercept. We also used the sympy library in Python to solve the equations and find the intercepts.

Example

Let's consider an example to illustrate the concept. Suppose we have the equation 2x + 3y - 5 = 0. To find the x-intercept, we set y = 0 and solve for x:

# Import necessary modules
import sympy as sp

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

# Substitute y = 0 into the equation
equation = 2*x + 3*y - 5
equation = equation.subs(y, 0)

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

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

This will output the x-intercept as -5/2.

Similarly, to find the y-intercept, we set x = 0 and solve for y:

# Import necessary modules
import sympy as sp

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

# Substitute x = 0 into the equation
equation = 2*x + 3*y - 5
equation = equation.subs(x, 0)

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

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

This will output the y-intercept as -5/3.

Applications

The concept of intercepts has numerous applications in mathematics and science. For example, in physics, the intercepts of a linear equation can be used to represent the position and velocity of an object. In engineering, the intercepts of a linear equation can be used to design and optimize systems.

Conclusion

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

A: The x-intercept of a linear equation is the point where the line crosses the x-axis. It is the value of x when y is equal to 0.

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

A: To find the x-intercept, set y = 0 and solve for x. You can use the sympy library in Python to solve the equation.

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

A: The y-intercept of a linear equation is the point where the line crosses the y-axis. It is the value of y when x is equal to 0.

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

A: To find the y-intercept, set x = 0 and solve for y. You can use the sympy library in Python to solve the equation.

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

A: 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. The x-intercept is the value of x when y is equal to 0, and the y-intercept is the value of y when x is equal to 0.

Q: Can I find the x- and y-intercepts of a linear equation using a graphing calculator?

A: Yes, you can find the x- and y-intercepts of a linear equation using a graphing calculator. Simply graph the equation and use the calculator to find the points where the line crosses the x-axis and the y-axis.

Q: Can I find the x- and y-intercepts of a linear equation using a computer algebra system (CAS)?

A: Yes, you can find the x- and y-intercepts of a linear equation using a computer algebra system (CAS). Simply enter the equation into the CAS and use the software to find the intercepts.

Q: What is the significance of the x- and y-intercepts of a linear equation?

A: The x- and y-intercepts of a linear equation are important because they represent the points where the line crosses the x-axis and the y-axis. These points are useful for graphing the equation and for solving problems involving the equation.

Q: Can I use the x- and y-intercepts of a linear equation to solve problems in other areas of mathematics?

A: Yes, you can use the x- and y-intercepts of a linear equation to solve problems in other areas of mathematics. For example, you can use the intercepts to solve systems of linear equations, to find the equation of a line passing through two points, and to solve problems involving linear inequalities.

Q: Can I use the x- and y-intercepts of a linear equation to solve problems in science and engineering?

A: Yes, you can use the x- and y-intercepts of a linear equation to solve problems in science and engineering. For example, you can use the intercepts to model the motion of an object, to find the equation of a line representing a physical phenomenon, and to solve problems involving linear relationships in science and engineering.

Q: What are some common mistakes to avoid when finding the x- and y-intercepts of a linear equation?

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

Failing to set y = 0 to find the x-intercept
Failing to set x = 0 to find the y-intercept
Not using the correct equation to find the intercepts
Not checking the work to ensure that the intercepts are correct

Q: How can I practice finding the x- and y-intercepts of a linear equation?

A: You can practice finding the x- and y-intercepts of a linear equation by working through examples and exercises in a textbook or online resource. You can also use a graphing calculator or computer algebra system (CAS) to practice finding the intercepts.