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Chapter 11: Problem 162

In the following exercises, graph using the intercepts. $$ x+y=-1 $$

Short Answer

Expert verified
The x-intercept is -1; the y-intercept is -1. Plot these and draw the line.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation. Solve for x. \[ x + 0 = -1 \] So, the x-intercept is \( x = -1 \). The point is (-1, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation. Solve for y. \[ 0 + y = -1 \] So, the y-intercept is \( y = -1 \). The point is (0, -1).
03

Plot the intercepts on a graph

Plot the points (-1, 0) and (0, -1) on a coordinate plane. These are the intercepts.
04

Draw the line

Draw a straight line through the points (-1, 0) and (0, -1). This line represents the equation \( x + y = -1 \).

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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 where the graph of an equation crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept from an equation, set y to zero and solve for x. For example, in the equation \(x+y=-1\), set y to 0 and solve for x: \[x+0=-1\]. This gives the x-intercept as \( x = -1 \). The coordinate for this point is (-1, 0). It's important to understand that the x-intercept tells us where the line cuts across the x-axis, providing one point to help us graph the equation.
y-intercept
The y-intercept is where the graph of an equation crosses the y-axis. In this case, the value of x is always zero. To find the y-intercept from an equation, set x to zero and solve for y. Using the equation \(x+y=-1\) again, we set x to 0: \[0 + y = -1 \]. This gives the y-intercept as \( y = -1 \). The coordinate for this point is (0, -1). The y-intercept is crucial because it shows us where the line intersects the y-axis, giving us another point to plot the graph accurately.
coordinate plane
The coordinate plane is a two-dimensional surface. It is formed by two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin (0, 0). The plane is divided into four quadrants. Each point on the coordinate plane is represented by a pair of numbers (x, y), known as coordinates. These coordinates tell us the position of the point relative to the origin. For example, the point (-1, 0) is 1 unit to the left of the origin on the x-axis and on the y-axis, while the point (0, -1) is exactly 1 unit down from the origin on the y-axis.
linear equations
Linear equations are equations of the form \(Ax + By = C\), where A, B, and C are constants. They graph as straight lines. The equation \(x + y = -1\) is a linear equation. When graphing linear equations, finding the x-intercept and y-intercept can simplify plotting the line. Follow these steps for any linear equation:
  • Find the x-intercept by setting y to zero and solving for x.
  • Find the y-intercept by setting x to zero and solving for y.
  • Plot these points on the coordinate plane.
  • Draw a line through these points.
This method ensures the line accurately represents the equation and helps visualize solutions easily.

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