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Chapter 11: Problem 159

In the following exercises, graph using the intercepts. $$ x+y=2 $$

Short Answer

Expert verified
The x-intercept is (2, 0) and the y-intercept is (0, 2). Plot these points and draw a line through them.

Step by step solution

01

- Identify the Intercepts

To graph the equation using intercepts, first find the x-intercept and the y-intercept. The x-intercept is where the graph crosses the x-axis (y=0), and the y-intercept is where it crosses the y-axis (x=0).
02

- Calculate the x-Intercept

Set y to 0 in the equation and solve for x: \[ x + y = 2 \]Set y to 0: \[ x + 0 = 2 \]So, the x-intercept is: \[ x = 2 \]Thus, the x-intercept is (2, 0).
03

- Calculate the y-Intercept

Set x to 0 in the equation and solve for y: \[ x + y = 2 \]Set x to 0: \[ 0 + y = 2 \]So, the y-intercept is: \[ y = 2 \]Thus, the y-intercept is (0, 2).
04

- Plot the Intercepts

On a graph, plot the points (2, 0) and (0, 2) which are the x-intercept and y-intercept respectively.
05

- Draw the Line

Draw a straight line passing through the points (2, 0) and (0, 2). This line is the graph of the equation \[ x + y = 2 \].

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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 line crosses the x-axis. At this point, the value of y is always zero.

To find the x-intercept for the equation \(x + y = 2\), set y to 0 and solve for x.

\r
    \r
  • Substitute y = 0 into the equation: \[ x + 0 = 2 \]
    • \r
  • Solve for x: \[ x = 2 \]
    • \r
The x-intercept is (2, 0).
Plot this point on the x-axis of a coordinate plane.
y-intercept
The y-intercept of a linear equation is where the line crosses the y-axis. Here, the value of x is always zero.

To find the y-intercept for the equation \(x + y = 2\), set x to 0 and solve for y.

\r
    \r
  • Substitute x = 0 into the equation: \[ 0 + y = 2 \]
    • \r
  • Solve for y: \[ y = 2 \]
    • \r
The y-intercept is (0, 2).
Plot this point on the y-axis of the coordinate plane.
linear equations
Linear equations are equations of the first degree, which means they have variables raised only to the power of one. These equations form straight lines when graphed on a coordinate plane.

The standard form of a linear equation is \(Ax + By = C\).

\r
    \r
  • In this form, A, B, and C are constants.
    • \r
  • x and y are variables.
    • \r
In this example, the linear equation is \(x + y = 2\), which makes both A and B equal to 1, and C equals 2.
This equation will produce a straight line when plotted on a graph.
coordinate plane
The coordinate plane is a two-dimensional surface used for graphing equations. It consists of two perpendicular lines:
\r
    \r
  • Horizontal line: x-axis
    • \r
  • Vertical line: y-axis
    • \r
The point where these axes intersect is called the origin, which has coordinates (0, 0).

To graph a linear equation, like \(x + y = 2\), plot the intercepts on the coordinate plane.

First, plot the x-intercept (2, 0) and the y-intercept (0, 2).
Next, draw a straight line passing through both points.
This line represents the graph of the equation \(x + y = 2\).

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