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Chapter 11: Problem 160

In the following exercises, graph using the intercepts. $$ x+y=5 $$

Short Answer

Expert verified
Plot the intercepts (5,0) and (0,5), then draw a line through them.

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 in the equation \[ x + y = 5 \] This becomes: \[ x + 0 = 5 \] Therefore, the x-intercept is \( x = 5 \). So, the point \((5, 0)\) is on the graph.
02

- Find the y-intercept

To find the y-intercept, set x to 0 in the equation \[ x + y = 5 \] This becomes: \[ 0 + y = 5 \] Therefore, the y-intercept is \( y = 5 \). So, the point \((0, 5)\) is on the graph.
03

- Plot the intercepts

Plot the points \((5,0)\) and \((0,5)\) on the coordinate plane. These points represent intercepts.
04

- Draw the line

Using a ruler, draw a straight line through the points \((5,0)\) and \((0,5)\). This is the graph of the equation \( x + y = 5 \). The line extends infinitely in both directions.

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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 on a graph. To find this point in any linear equation, you set the value of y to 0 and then solve for x. In other words, you're looking for the value of x when y is zero. For example, in the equation \[ x + y = 5 \], if you set y to 0, you get \[ x + 0 = 5 \], which simplifies to \[ x = 5 \]. Therefore, the x-intercept is at the point (5, 0). When graphing, this represents a plot on the horizontal axis. The x-intercept is a vital point to find as it simplifies the process of graphing linear equations.
y-intercept
The y-intercept is where a line crosses the y-axis. To find this point in any given linear equation, you set the value of x to 0 and solve for y. Essentially, this is finding the value of y when x is zero. For example, using the equation \[ x + y = 5 \], if you set x to 0, the equation becomes \[ 0 + y = 5 \], simplifying to \[ y = 5 \]. Therefore, the y-intercept is at the point (0, 5). This point lies on the vertical axis. Discovering the y-intercept helps in accurately plotting the line on a graph, giving you a clear starting point on the vertical axis.
coordinate plane
The coordinate plane is a two-dimensional surface where we can graphically represent equations and data. It is formed by two perpendicular lines called the x-axis and y-axis. These axes intersect at a point called the origin, which is labeled as (0, 0). The plane is divided into four quadrants:
  • Quadrant I: Positive x and y values
  • Quadrant II: Negative x and positive y values
  • Quadrant III: Negative x and y values
  • Quadrant IV: Positive x and negative y values
Each point on the coordinate plane is represented by a pair of numbers (x, y), indicating its position relative to the origin. Understanding the coordinate plane is crucial for graphing equations and knowing where points will lie.
plotting points
Plotting points on a coordinate plane involves locating and marking specific pairs of (x, y) values. Here's how to plot points:
  • Identify the point you need to plot.
  • Start at the origin (0, 0) on the coordinate plane.
  • Move horizontally to the x-value of the point.
  • From there, move vertically to the y-value.
  • Mark the point where this final position is.
For example, to plot the point (5, 0), you move 5 units to the right along the x-axis and then stay at 0 on the y-axis. For the point (0, 5), you stay at 0 on the x-axis and move 5 units up the y-axis. Once the points are plotted, you can draw a line through them to represent the equation graphically. Plotting points correctly ensures that the equation is accurately represented on the graph.

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Most popular questions from this chapter

In the following exercises, identify the most convenient method to graph each line. $$ x-y=5 $$

In the following exercises, find three solutions to each linear equation. $$ y=3 x+1 $$

In the following exercises, find the intercepts. $$ y=\frac{3}{4} x-12 $$

Is \((1,3)\) a solution to the equation \(x+4 y=12 ?\) How do you know?

In the following exercises, identify the most convenient method to graph each line. $$ y=\frac{2}{3} x-1 $$
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