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Chapter 4: Problem 155

Find the intercepts for each equation. \(x-y=5\)

Short Answer

Expert verified
The x-intercept is (5, 0) and the y-intercept is (0, -5).

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y = 0 in the equation and solve for x. Substituting y = 0, the equation becomes: x - 0 = 5 This simplifies to: x = 5 Therefore, the x-intercept is (5, 0).
02

- Find the y-intercept

To find the y-intercept, set x = 0 in the equation and solve for y. Substituting x = 0, the equation becomes: 0 - y = 5 This simplifies to: -y = 5 Dividing both sides by -1 gives: y = -5 Therefore, the y-intercept is (0, -5).

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Key Concepts

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

x-intercept
The x-intercept is the point where a line crosses the x-axis. This happens when the y-coordinate of the point is zero. To find the x-intercept for an equation, you set y = 0 and solve for x. For example, in the equation \(x - y = 5\), if we set \(y = 0\), it becomes \(x - 0 = 5\). Simplifying this, we get \(x = 5\). Therefore, the x-intercept is at the point (5, 0). Here鈥檚 a quick recap:
  • Set y = 0.
  • Substitute into the equation.
  • Solve for x.
  • The result is your x-intercept.
Understanding this fundamental concept is crucial as it helps pinpoint where the graph of a linear equation crosses the x-axis.
y-intercept
The y-intercept is the point where a line crosses the y-axis. This occurs when the x-coordinate of the point is zero. To find the y-intercept for an equation, you set x = 0 and solve for y. Let's use the same equation: \(x - y = 5\). If we set \(x = 0\), the equation changes to \(0 - y = 5\). Simplifying this, we get \(-y = 5\). Dividing both sides by -1 gives us \(y = -5\), so the y-intercept is at (0, -5). Remember these steps:
  • Set x = 0.
  • Substitute into the equation.
  • Solve for y.
  • This value is your y-intercept.
The y-intercept helps determine where the graph of a linear equation crosses the y-axis.
solving linear equations
Solving linear equations involves finding the values of the variables that make the equation true. A linear equation typically takes the form \(ax + by = c\), where a, b, and c are constants. This equation can be solved for one variable if the other is known. Here are the key steps:
  • Isolate the variable: Move terms involving the variable you are solving for to one side of the equation and the constant terms to the other.
  • Simplify the equation: Combine like terms and simplify both sides of the equation as much as possible.
  • Solve for the variable: Perform algebraic operations to solve for the variable.

In our example with \(x - y = 5\), we showcased isolating the variables by simplifying the equation when setting \(y = 0\) for x-intercept or \(x = 0\) for y-intercept. Understanding how to solve these equations is key to mastering algebra concepts and graphing linear functions.

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Most popular questions from this chapter

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