Find The { X $}$-intercept And { Y $} − I N T E R C E P T O F T H E L I N E . -intercept Of The Line. − In T Erce Pt O F T H E L In E . { 2x + 6y = -18 \}
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Introduction
In mathematics, the x-intercept and y-intercept of a line are the points at which the line crosses the x-axis and y-axis, respectively. These intercepts are crucial in understanding the behavior and properties of a line. In this article, we will focus on finding the x- and y-intercepts of a linear equation in the form of 2x + 6y = -18.
What are x- and y-intercepts?
The x-intercept of a line is the point at which the line crosses the x-axis. At this point, the value of y is always 0. Similarly, the y-intercept of a line is the point at which the line crosses the y-axis. At this point, the value of x is always 0.
Finding the x-intercept
To find the x-intercept of the line 2x + 6y = -18, we need to substitute y = 0 into the equation and solve for x.
Step 1: Substitute y = 0 into the equation
2x + 6(0) = -18
Step 2: Simplify the equation
2x = -18
Step 3: Solve for x
x = -18/2 x = -9
Therefore, the x-intercept of the line 2x + 6y = -18 is (-9, 0).
Finding the y-intercept
To find the y-intercept of the line 2x + 6y = -18, we need to substitute x = 0 into the equation and solve for y.
Step 1: Substitute x = 0 into the equation
2(0) + 6y = -18
Step 2: Simplify the equation
6y = -18
Step 3: Solve for y
y = -18/6 y = -3
Therefore, the y-intercept of the line 2x + 6y = -18 is (0, -3).
Conclusion
In conclusion, finding the x- and y-intercepts of a linear equation is a crucial step in understanding the behavior and properties of a line. By substituting y = 0 and x = 0 into the equation, we can find the x-intercept and y-intercept, respectively. In this article, we have found the x- and y-intercepts of the line 2x + 6y = -18 to be (-9, 0) and (0, -3), respectively.
Example Problems
Problem 1
Find the x- and y-intercepts of the line 3x + 2y = 12.
Solution
To find the x-intercept, substitute y = 0 into the equation and solve for x. 3x + 2(0) = 12 3x = 12 x = 12/3 x = 4
To find the y-intercept, substitute x = 0 into the equation and solve for y. 3(0) + 2y = 12 2y = 12 y = 12/2 y = 6
Therefore, the x-intercept and y-intercept of the line 3x + 2y = 12 are (4, 0) and (0, 6), respectively.
Problem 2
Find the x- and y-intercepts of the line x + 4y = 10.
Solution
To find the x-intercept, substitute y = 0 into the equation and solve for x. x + 4(0) = 10 x = 10
To find the y-intercept, substitute x = 0 into the equation and solve for y. 0 + 4y = 10 4y = 10 y = 10/4 y = 2.5
Therefore, the x-intercept and y-intercept of the line x + 4y = 10 are (10, 0) and (0, 2.5), respectively.
Applications
Finding the x- and y-intercepts of a linear equation has numerous applications in various fields, including:
- Graphing: Finding the x- and y-intercepts is essential in graphing a line on a coordinate plane.
- Solving Systems of Equations: Finding the x- and y-intercepts can help in solving systems of linear equations.
- Linear Programming: Finding the x- and y-intercepts is crucial in linear programming, where it is used to find the optimal solution to a problem.
- Physics and Engineering: Finding the x- and y-intercepts is used in physics and engineering to model real-world problems and find the optimal solution.
Final Thoughts
In conclusion, finding the x- and y-intercepts of a linear equation is a fundamental concept in mathematics. By understanding how to find these intercepts, we can apply this knowledge to various fields and solve real-world problems.
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Q: What is the x-intercept of a line?
A: The x-intercept of a line is the point at which the line crosses the x-axis. At this point, the value of y is always 0.
Q: How do I find the x-intercept of a line?
A: To find the x-intercept of a line, substitute y = 0 into the equation and solve for x.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point at which the line crosses the y-axis. At this point, the value of x is always 0.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept of a line, substitute x = 0 into the equation and solve for y.
Q: Can I find the x- and y-intercepts of a line using a graph?
A: Yes, you can find the x- and y-intercepts of a line using a graph. 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.
Q: What is the difference between the x-intercept and the 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. The x-intercept has a y-coordinate of 0, and the y-intercept has an x-coordinate of 0.
Q: Can I find the x- and y-intercepts of a line with a negative slope?
A: Yes, you can find the x- and y-intercepts of a line with a negative slope. The process is the same as finding the intercepts of a line with a positive slope.
Q: Can I find the x- and y-intercepts of a line with a zero slope?
A: Yes, you can find the x- and y-intercepts of a line with a zero slope. The line will be a horizontal line, and the x- and y-intercepts will be the same point.
Q: Can I find the x- and y-intercepts of a line with a vertical slope?
A: Yes, you can find the x- and y-intercepts of a line with a vertical slope. The line will be a vertical line, and the x- and y-intercepts will be the same point.
Q: What is the significance of finding the x- and y-intercepts of a line?
A: Finding the x- and y-intercepts of a line is significant because it helps in understanding the behavior and properties of the line. It also helps in graphing the line on a coordinate plane and solving systems of linear equations.
Q: Can I use a calculator to find the x- and y-intercepts of a line?
A: Yes, you can use a calculator to find the x- and y-intercepts of a line. Most graphing calculators have a built-in function to find the intercepts of a line.
Q: Can I find the x- and y-intercepts of a line with a fractional slope?
A: Yes, you can find the x- and y-intercepts of a line with a fractional slope. The process is the same as finding the intercepts of a line with a whole number slope.
Q: Can I find the x- and y-intercepts of a line with a negative fractional slope?
A: Yes, you can find the x- and y-intercepts of a line with a negative fractional slope. The process is the same as finding the intercepts of a line with a positive fractional slope.
Q: Can I find the x- and y-intercepts of a line with a zero fractional slope?
A: Yes, you can find the x- and y-intercepts of a line with a zero fractional slope. The line will be a horizontal line, and the x- and y-intercepts will be the same point.
Q: Can I find the x- and y-intercepts of a line with a vertical fractional slope?
A: Yes, you can find the x- and y-intercepts of a line with a vertical fractional slope. The line will be a vertical line, and the x- and y-intercepts will be the same point.
Q: What are some real-world applications of finding the x- and y-intercepts of a line?
A: Some real-world applications of finding the x- and y-intercepts of a line include:
- Graphing: Finding the x- and y-intercepts is essential in graphing a line on a coordinate plane.
- Solving Systems of Equations: Finding the x- and y-intercepts can help in solving systems of linear equations.
- Linear Programming: Finding the x- and y-intercepts is crucial in linear programming, where it is used to find the optimal solution to a problem.
- Physics and Engineering: Finding the x- and y-intercepts is used in physics and engineering to model real-world problems and find the optimal solution.
Q: Can I use the x- and y-intercepts to solve a system of linear equations?
A: Yes, you can use the x- and y-intercepts to solve a system of linear equations. By finding the intercepts, you can determine the point of intersection of the two lines.
Q: Can I use the x- and y-intercepts to find the equation of a line?
A: Yes, you can use the x- and y-intercepts to find the equation of a line. By using the intercepts, you can determine the slope and y-intercept of the line.
Q: Can I use the x- and y-intercepts to graph a line?
A: Yes, you can use the x- and y-intercepts to graph a line. By plotting the intercepts on a coordinate plane, you can draw the line.
Q: Can I use the x- and y-intercepts to solve a linear programming problem?
A: Yes, you can use the x- and y-intercepts to solve a linear programming problem. By finding the intercepts, you can determine the optimal solution to the problem.
Q: Can I use the x- and y-intercepts to model real-world problems in physics and engineering?
A: Yes, you can use the x- and y-intercepts to model real-world problems in physics and engineering. By finding the intercepts, you can determine the optimal solution to the problem.
Q: Can I use the x- and y-intercepts to solve a system of linear inequalities?
A: Yes, you can use the x- and y-intercepts to solve a system of linear inequalities. By finding the intercepts, you can determine the feasible region of the system.
Q: Can I use the x- and y-intercepts to find the equation of a circle?
A: Yes, you can use the x- and y-intercepts to find the equation of a circle. By using the intercepts, you can determine the center and radius of the circle.
Q: Can I use the x- and y-intercepts to graph a circle?
A: Yes, you can use the x- and y-intercepts to graph a circle. By plotting the intercepts on a coordinate plane, you can draw the circle.
Q: Can I use the x- and y-intercepts to solve a system of nonlinear equations?
A: Yes, you can use the x- and y-intercepts to solve a system of nonlinear equations. By finding the intercepts, you can determine the point of intersection of the two curves.
Q: Can I use the x- and y-intercepts to find the equation of a nonlinear curve?
A: Yes, you can use the x- and y-intercepts to find the equation of a nonlinear curve. By using the intercepts, you can determine the shape and position of the curve.
Q: Can I use the x- and y-intercepts to graph a nonlinear curve?
A: Yes, you can use the x- and y-intercepts to graph a nonlinear curve. By plotting the intercepts on a coordinate plane, you can draw the curve.
Q: Can I use the x- and y-intercepts to solve a system of parametric equations?
A: Yes, you can use the x- and y-intercepts to solve a system of parametric equations. By finding the intercepts, you can determine the point of intersection of the two curves.
Q: Can I use the x- and y-intercepts to find the equation of a parametric curve?
A: Yes, you can use the x- and y-intercepts to find the equation of a parametric curve. By using the intercepts, you can determine the shape and position of the curve.
Q: Can I use the x- and y-intercepts to graph a parametric curve?
A: Yes, you can use the x- and y-intercepts to graph a parametric curve. By plotting the intercepts on a coordinate plane, you can draw the curve.
Q: Can I use the x- and y-intercepts to solve a system of polar equations?
A: Yes, you can use the x- and y-intercepts to solve a system of polar equations. By finding the intercepts, you can determine the point of intersection of the two curves.