Find The { X $}$- And { Y $} − I N T E R C E P T S F O R T H E E Q U A T I O N : -intercepts For The Equation: − In T Erce Pt S F Or T H Ee Q U A T I O N : { Y = \frac{x-3}{2} \} - { X $}$-intercept: { (x, Y) = (\square, 0)$}$- { Y $}$-intercept: { (x, Y) = (0, \square)$}$

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Understanding the Concept of Intercepts

In mathematics, the intercepts of an equation are the points at which the graph of the equation intersects the x-axis and the y-axis. The x-intercept is the point at which the graph intersects the x-axis, and the y-intercept is the point at which the graph intersects the y-axis. In this article, we will focus on finding the x-intercept and the y-intercept for the given equation.

The Given Equation

The given equation is:

y=x−32{ y = \frac{x-3}{2} }

This is a linear equation in the form of y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is 1/2 and the y-intercept is -3/2.

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 is the point at which the graph intersects the x-axis, and at this point, the value of y is 0.

Setting y = 0, we get:

0=x−32{ 0 = \frac{x-3}{2} }

Multiplying both sides by 2, we get:

0=x−3{ 0 = x - 3 }

Adding 3 to both sides, we get:

3=x{ 3 = x }

Therefore, the x-intercept is (3, 0).

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 is the point at which the graph intersects the y-axis, and at this point, the value of x is 0.

Setting x = 0, we get:

y=0−32{ y = \frac{0-3}{2} }

Simplifying, we get:

y=−32{ y = -\frac{3}{2} }

Therefore, the y-intercept is (0, -3/2).

Conclusion

In this article, we have found the x-intercept and the y-intercept for the given equation. The x-intercept is (3, 0) and the y-intercept is (0, -3/2). These intercepts are important because they provide information about the graph of the equation and can be used to solve problems involving the equation.

Example Problems

Problem 1

Find the x-intercept and the y-intercept for the equation:

y=x+23{ y = \frac{x+2}{3} }

Solution

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

0=x+23{ 0 = \frac{x+2}{3} }

Multiplying both sides by 3, we get:

0=x+2{ 0 = x + 2 }

Subtracting 2 from both sides, we get:

−2=x{ -2 = x }

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

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

y=0+23{ y = \frac{0+2}{3} }

Simplifying, we get:

y=23{ y = \frac{2}{3} }

Therefore, the y-intercept is (0, 2/3).

Problem 2

Find the x-intercept and the y-intercept for the equation:

y=x−14{ y = \frac{x-1}{4} }

Solution

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

0=x−14{ 0 = \frac{x-1}{4} }

Multiplying both sides by 4, we get:

0=x−1{ 0 = x - 1 }

Adding 1 to both sides, we get:

1=x{ 1 = x }

Therefore, the x-intercept is (1, 0).

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

y=0−14{ y = \frac{0-1}{4} }

Simplifying, we get:

y=−14{ y = -\frac{1}{4} }

Therefore, the y-intercept is (0, -1/4).

Applications of Intercepts

Intercepts have many applications in mathematics and real-world problems. Some of the applications include:

  • Graphing: Intercepts are used to graph linear equations. By finding the x-intercept and the y-intercept, we can plot the graph of the equation.
  • Solving Systems of Equations: Intercepts are used to solve systems of linear equations. By finding the x-intercept and the y-intercept of each equation, we can solve the system of equations.
  • Real-World Problems: Intercepts are used to solve real-world problems involving linear equations. For example, finding the x-intercept and the y-intercept of an equation can help us solve problems involving cost, revenue, and profit.

Conclusion

In this article, we have discussed the concept of intercepts and how to find the x-intercept and the y-intercept for a given equation. We have also provided example problems and discussed the applications of intercepts in mathematics and real-world problems.

Understanding the Concept of Intercepts

In mathematics, the intercepts of an equation are the points at which the graph of the equation intersects the x-axis and the y-axis. The x-intercept is the point at which the graph intersects the x-axis, and the y-intercept is the point at which the graph intersects the y-axis. In this article, we will focus on finding the x-intercept and the y-intercept for the given equation.

Q&A

Q: What is the x-intercept of the equation y = (x-3)/2?

A: To find the x-intercept, we need to set y = 0 and solve for x. This is because the x-intercept is the point at which the graph intersects the x-axis, and at this point, the value of y is 0.

Setting y = 0, we get:

0=x−32{ 0 = \frac{x-3}{2} }

Multiplying both sides by 2, we get:

0=x−3{ 0 = x - 3 }

Adding 3 to both sides, we get:

3=x{ 3 = x }

Therefore, the x-intercept is (3, 0).

Q: What is the y-intercept of the equation y = (x-3)/2?

A: To find the y-intercept, we need to set x = 0 and solve for y. This is because the y-intercept is the point at which the graph intersects the y-axis, and at this point, the value of x is 0.

Setting x = 0, we get:

y=0−32{ y = \frac{0-3}{2} }

Simplifying, we get:

y=−32{ y = -\frac{3}{2} }

Therefore, the y-intercept is (0, -3/2).

Q: How do I find the x-intercept and the y-intercept for a given equation?

A: To find the x-intercept and the y-intercept for a given equation, you need to follow these steps:

  • Set y = 0 and solve for x to find the x-intercept.
  • Set x = 0 and solve for y to find the y-intercept.

Q: What are the applications of intercepts in mathematics and real-world problems?

A: Intercepts have many applications in mathematics and real-world problems. Some of the applications include:

  • Graphing: Intercepts are used to graph linear equations. By finding the x-intercept and the y-intercept, we can plot the graph of the equation.
  • Solving Systems of Equations: Intercepts are used to solve systems of linear equations. By finding the x-intercept and the y-intercept of each equation, we can solve the system of equations.
  • Real-World Problems: Intercepts are used to solve real-world problems involving linear equations. For example, finding the x-intercept and the y-intercept of an equation can help us solve problems involving cost, revenue, and profit.

Q: Can I use the intercepts to solve systems of equations?

A: Yes, you can use the intercepts to solve systems of equations. By finding the x-intercept and the y-intercept of each equation, you can solve the system of equations.

Q: How do I use the intercepts to solve real-world problems?

A: To use the intercepts to solve real-world problems, you need to follow these steps:

  • Identify the equation that represents the problem.
  • Find the x-intercept and the y-intercept of the equation.
  • Use the intercepts to solve the problem.

Conclusion

In this article, we have discussed the concept of intercepts and how to find the x-intercept and the y-intercept for a given equation. We have also provided example problems and discussed the applications of intercepts in mathematics and real-world problems.