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Chapter 11: Problem 130

In the following exercises, find the intercepts. $$ x+y=-5 $$

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. The original equation is: \( x + y = -5 \)Setting y = 0:\( x + 0 = -5 \)So, \( x = -5 \)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. Using the original equation again: \( x + y = -5 \)Setting x = 0:\( 0 + y = -5 \)So, \( y = -5 \)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 of a linear equation is the point where the graph of the equation crosses the x-axis. To find the x-intercept, we set the value of y to zero and solve for x. This gives us the coordinates of the x-intercept in the form \(x, 0\).

Let's take the equation given in the exercise: \( x + y = -5 \). To find the x-intercept:
  • Set y to 0 in the equation: \(x + 0 = -5\)
  • Solve for x: \(x = -5\)
Therefore, the x-intercept is \(-5, 0\). This indicates that the graph crosses the x-axis at \(-5\). By understanding this method, you can apply it to any linear equation to find the x-intercept.
y-intercept
The y-intercept of a linear equation is the point where the graph of the equation crosses the y-axis. To find the y-intercept, we set the value of x to zero and solve for y. This gives us the coordinates of the y-intercept in the form \(0, y\).

For the same equation provided: \( x + y = -5 \):
  • Set x to 0 in the equation: \( 0 + y = -5 \)
  • Solve for y: \( y = -5 \)
Thus, the y-intercept is \(0, -5\). This means that the graph crosses the y-axis at \(-5\). Understanding this technique helps you find the y-intercept for any linear equation.
linear equations
Linear equations are equations that represent straight lines when graphed on a coordinate plane. They generally follow the form \(Ax + By = C\), where A, B, and C are constants.

One key characteristic of linear equations is that their graphs produce straight lines, which makes intercepts straightforward to find. By setting either variable (x or y) to zero, we can easily determine where the line crosses the axes (intercepts).
  • To find the x-intercept, set y to 0 and solve for x.
  • To find the y-intercept, set x to 0 and solve for y.
This exercise demonstrates these principles with the equation \(x + y = -5\). By understanding how to manipulate and solve linear equations, you gain insights into the relationship between algebraic expressions and their graphical representations.

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Most popular questions from this chapter

In the following exercises, graph by plotting points. $$ y=-3 x $$

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