Find The { Y $}$-intercept And { X $}$-intercept Of The Line. ${ X - 3y = 6 }$

by ADMIN 81 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 at which the line crosses the x-axis, and the y-intercept is the point at which the line crosses 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 of ax + by = c.

Understanding the Equation

The given equation is x - 3y = 6. To find the intercepts, we need to isolate the variables x and y. We can do this by rearranging the equation to solve for x and y.

Finding the y-intercept

The y-intercept is the point at which the line crosses the y-axis. This occurs when x = 0. To find the y-intercept, we can substitute x = 0 into the equation and solve for y.

x - 3y = 6
0 - 3y = 6
-3y = 6
y = -2

Therefore, the y-intercept is (-2, 0).

Finding the x-intercept

The x-intercept is the point at which the line crosses the x-axis. This occurs when y = 0. To find the x-intercept, we can substitute y = 0 into the equation and solve for x.

x - 3y = 6
x - 3(0) = 6
x = 6

Therefore, the x-intercept is (6, 0).

Conclusion

In this article, we discussed how to find the x-intercept and y-intercept of a linear equation in the form of ax + by = c. We used the given equation x - 3y = 6 to find the intercepts. The y-intercept is the point at which the line crosses the y-axis, and the x-intercept is the point at which the line crosses the x-axis. By following the steps outlined in this article, you can find the intercepts of any linear equation.

Example Problems

Problem 1

Find the x-intercept and y-intercept of the line 2x + 3y = 12.

Solution

To find the y-intercept, we can substitute x = 0 into the equation and solve for y.

2x + 3y = 12
2(0) + 3y = 12
3y = 12
y = 4

Therefore, the y-intercept is (0, 4).

To find the x-intercept, we can substitute y = 0 into the equation and solve for x.

2x + 3y = 12
2x + 3(0) = 12
2x = 12
x = 6

Therefore, the x-intercept is (6, 0).

Problem 2

Find the x-intercept and y-intercept of the line x - 2y = 4.

Solution

To find the y-intercept, we can substitute x = 0 into the equation and solve for y.

x - 2y = 4
0 - 2y = 4
-2y = 4
y = -2

Therefore, the y-intercept is (-2, 0).

To find the x-intercept, we can substitute y = 0 into the equation and solve for x.

x - 2y = 4
x - 2(0) = 4
x = 4

Therefore, the x-intercept is (4, 0).

Tips and Tricks

  • To find the y-intercept, substitute x = 0 into the equation and solve for y.
  • To find the x-intercept, substitute y = 0 into the equation and solve for x.
  • Make sure to isolate the variables x and y before solving for the intercepts.
  • Use the given equation to find the intercepts, and do not use any additional information.

Common Mistakes

  • Failing to isolate the variables x and y before solving for the intercepts.
  • Using the wrong equation to find the intercepts.
  • Not checking the work for errors.

Real-World Applications

  • Finding the intercepts of a linear equation is used in many real-world applications, such as:
  • Graphing linear equations
  • Finding the equation of a line
  • Solving systems of linear equations
  • Calculating the slope of a line

Conclusion

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

A: The x-intercept of a linear equation is the point at which the line crosses the x-axis. This occurs when y = 0.

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

A: The y-intercept of a linear equation is the point at which the line crosses the y-axis. This occurs when x = 0.

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

A: To find the x-intercept, substitute y = 0 into the equation and solve for x.

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

A: To find the y-intercept, substitute x = 0 into the equation and solve for y.

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

A: The x-intercept is the point at which the line crosses the x-axis, and the y-intercept is the point at which the line crosses the y-axis.

Q: Can I find the intercepts of a linear equation using a graph?

A: Yes, you can find the intercepts of a linear equation using a graph. The x-intercept is the point at which the line crosses the x-axis, and the y-intercept is the point at which the line crosses the y-axis.

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 find the slope of a line?

A: To find the slope of a line, use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: Can I find the intercepts of a linear equation using a calculator?

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

Q: What is the significance of the intercepts of a linear equation?

A: The intercepts of a linear equation are important because they help us understand the behavior of the line. The x-intercept and y-intercept can be used to graph the line and to find the equation of the line.

Q: Can I find the intercepts of a linear equation using a graphing software?

A: Yes, you can find the intercepts of a linear equation using a graphing software. Most graphing software, such as Desmos or GeoGebra, have a built-in function to find the x-intercept and y-intercept of a linear equation.

Q: What is the difference between the x-intercept and the y-intercept in a quadratic equation?

A: In a quadratic equation, the x-intercept is the point at which the parabola crosses the x-axis, and the y-intercept is the point at which the parabola crosses the y-axis.

Q: Can I find the intercepts of a quadratic equation using a graph?

A: Yes, you can find the intercepts of a quadratic equation using a graph. The x-intercept is the point at which the parabola crosses the x-axis, and the y-intercept is the point at which the parabola crosses the y-axis.

Q: What is the significance of the intercepts of a quadratic equation?

A: The intercepts of a quadratic equation are important because they help us understand the behavior of the parabola. The x-intercept and y-intercept can be used to graph the parabola and to find the equation of the parabola.

Q: Can I find the intercepts of a quadratic equation using a calculator?

A: Yes, you can find the intercepts of a quadratic equation using a calculator. Most calculators have a built-in function to find the x-intercept and y-intercept of a quadratic equation.

Q: What is the difference between the x-intercept and the y-intercept in a cubic equation?

A: In a cubic equation, the x-intercept is the point at which the cubic curve crosses the x-axis, and the y-intercept is the point at which the cubic curve crosses the y-axis.

Q: Can I find the intercepts of a cubic equation using a graph?

A: Yes, you can find the intercepts of a cubic equation using a graph. The x-intercept is the point at which the cubic curve crosses the x-axis, and the y-intercept is the point at which the cubic curve crosses the y-axis.

Q: What is the significance of the intercepts of a cubic equation?

A: The intercepts of a cubic equation are important because they help us understand the behavior of the cubic curve. The x-intercept and y-intercept can be used to graph the cubic curve and to find the equation of the cubic curve.

Q: Can I find the intercepts of a cubic equation using a calculator?

A: Yes, you can find the intercepts of a cubic equation using a calculator. Most calculators have a built-in function to find the x-intercept and y-intercept of a cubic equation.