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Chapter 9: Problem 16

Determine the intercepts and graph each linear equation. $$x+y-2=0$$

Short Answer

Expert verified
The x-intercept is (2, 0) and the y-intercept is (0, 2).

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Given equation: \[ x + y - 2 = 0 \]Substitute y = 0: \[ x + 0 - 2 = 0 \]This simplifies to: \[ x = 2 \]So, the x-intercept is (2, 0).
02

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Given equation: \[ x + y - 2 = 0 \]Substitute x = 0: \[ 0 + y - 2 = 0 \]This simplifies to: \[ y = 2 \]So, the y-intercept is (0, 2).
03

- Plot the intercepts

Plot the x-intercept (2, 0) and the y-intercept (0, 2) on a coordinate plane.
04

- Draw the line through the intercepts

Draw a straight line through the points (2, 0) and (0, 2) to complete the graph of the linear equation.

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Key Concepts

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

x-intercept
To understand the x-intercept, start by setting y to 0 in your equation. The x-intercept is where the line crosses the x-axis. For the equation given, \( x + y - 2 = 0 \), you substitute y with 0, simplifying to \( x = 2 \). Therefore, the x-intercept is the coordinate (2, 0). Think of it as the point where your line touches the x-axis, providing a clear starting point for graphing.
y-intercept
Finding the y-intercept involves setting x to 0. It's the point where the line crosses the y-axis. For \( x + y - 2 = 0 \), plugging in x = 0 simplifies the equation to \( y = 2 \). So, the y-intercept here is (0, 2). Just like the x-intercept, this point is integral for sketching the graph as it tells you where the line meets the y-axis.
coordinate plane
The coordinate plane is a two-dimensional space formed by the intersection of two number lines: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane represents a pair of values (x, y). The x-coordinate shows the horizontal position, whereas the y-coordinate shows the vertical position. In our example, the intercepts (2, 0) and (0, 2) can be plotted on this plane to visualize the linear equation.
linear equations
Linear equations like \( x + y - 2 = 0 \) graph as straight lines. These equations follow the structure \( ax + by + c = 0 \), where a, b, and c are constants. The solutions to these equations are all the points forming the line. To graph, find the intercepts first. Plot them, then draw a straight line through these points. This line represents all potential solutions to the linear equation and gives a visual understanding of their relationships.

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