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Chapter 3: Problem 37

In the following exercises, find the intercepts for each equation. $$ x-y=5 $$

Short Answer

Expert verified
The intercepts are (5, 0) and (0, -5).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. The equation is \[ x - y = 5 \] Substitute y = 0: \[ x - 0 = 5 \] So, \[ x = 5 \] The x-intercept is (5, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. The equation is \[ x - y = 5 \] Substitute x = 0: \[ 0 - y = 5 \] So, \[ -y = 5 \] Therefore, \[ y = -5 \] The y-intercept is (0, -5).
03

Conclusion

The intercepts for the equation \( x - y = 5 \) are at (5, 0) for the x-intercept and (0, -5) for the y-intercept.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the x-intercept
The x-intercept is where the graph of an equation crosses the x-axis. At this point, the value of y is always 0 because you're on the x-axis. To find the x-intercept, you need to set y to 0 and solve for x. Let's consider the linear equation given in the exercise: \( x - y = 5 \). Set y to 0: \( x - 0 = 5 \) This simplifies to: \( x = 5 \) So, the x-intercept for this equation is (5, 0). Notice how simple it is to find the x-intercept once you set y to 0. Follow this method for any linear equation, and you'll always find the x-intercept.
Finding the y-intercept
The y-intercept is where the graph crosses the y-axis. At this point, the value of x is always 0 because you're on the y-axis. To find the y-intercept, you set x to 0 and solve for y. Using our given equation: \( x - y = 5 \). Set x to 0: \( 0 - y = 5 \) This simplifies to: \( -y = 5 \) Therefore, solving for y, we get: \( y = -5 \) Thus, the y-intercept for this equation is (0, -5). By setting x to 0, you can follow this method to find the y-intercept for any linear equation.
Linear Equations Explained
A linear equation is an equation whose graph forms a straight line. In its simplest form, a linear equation can be written as \( Ax + By = C \). Here \( A \), \( B \), and \( C \) are constants. The equation given, \( x - y = 5 \), is an example of a linear equation. Here's what to understand about linear equations:
  • They always graph to a straight line.
  • Each solution pair \( (x, y) \) lies on the line.
  • The x-intercept is found by setting y to 0.
  • The y-intercept is found by setting x to 0.
Linear equations are fundamental in algebra. They help us understand relationships between variables and model real-life situations. By mastering linear equations, you'll have a solid foundation for learning more complex mathematical concepts.

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