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Chapter 0: Problem 7

Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") \(x+y=5\)

Short Answer

Expert verified
Graph the line by plotting points for \(y = -x + 5\) and connect them.

Step by step solution

01

Understand the Equation

The given equation is a linear equation in two variables: \(x\) and \(y\). We are tasked with graphing this line on a coordinate plane.
02

Identify Key Components

The equation \(x + y = 5\) can be rewritten in slope-intercept form, \(y = -x + 5\). This tells us the slope of the line is \(-1\) and the y-intercept is \(5\).
03

Plot the Y-intercept

Start graphing by plotting the y-intercept. According to our equation \(y = -x + 5\), the y-intercept is \(5\), so place a point at \((0, 5)\) on the y-axis.
04

Use the Slope to Find Another Point

The slope of \(-1\) means that for each unit you move to the right along the x-axis, you move one unit down along the y-axis. From \((0, 5)\), move 1 unit to the right to \(x=1\), and 1 unit down to \(y=4\), placing another point at \((1, 4)\).
05

Draw the Line

Connect the two points \((0, 5)\) and \((1, 4)\) with a straight line. Extend this line in both directions; this is the graph of the equation \(x + y = 5\).
06

Verify the Line

To ensure accuracy, check additional points along the line by substituting values into the original equation. For example, substitute \(x=2\) into the equation to get \(y=3\) (since \(2 + 3 = 5\)). Verify this point \((2, 3)\) lies on the line. Repeat as needed to confirm the line is correct.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as \( y = mx + b \). This is a really handy format because it tells us two important things right away: the slope \( m \) and the y-intercept \( b \).

To convert an equation like \( x + y = 5 \) into slope-intercept form, we need to solve for \( y \). By rearranging, we get \( y = -x + 5 \). Here, \( m = -1 \), which represents the slope, and \( b = 5 \), which indicates where the line crosses the y-axis.

Understanding the slope-intercept form simplifies the process of graphing. Once you have this formula, plotting the y-intercept and using the slope to find other points on the line becomes a straightforward task.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It is defined by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).

Each point on the plane is described by a pair of numerical coordinates \((x, y)\). The x-coordinate indicates the position along the x-axis, and the y-coordinate shows the position along the y-axis.

When graphing a line like \( x + y = 5 \), it's essential to understand how these axes work. You plot points based on these coordinates and draw the line that passes through them. The coordinate plane is a crucial tool because it provides a visual representation of relationships between variables.
Y-Intercept
The y-intercept of a graph is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the \( b \) value is the y-intercept. For the equation \( y = -x + 5 \), the y-intercept is \( 5 \). This means the line crosses the y-axis at the point \((0, 5)\).

To begin graphing a linear equation, start by plotting the y-intercept. This point becomes your anchor on the coordinate plane. From this starting point, use the slope to find other points on the line.

This is why the y-intercept is so important: it gives you the first fixed position to start your graph. Whether you're new to graphing or looking for a reliable method, always begin at the y-intercept.

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Most popular questions from this chapter

A function \(h\) takes a number \(x\), subtracts \(4,\) and then squares the result, while a function \(k\) takes a number \(x\), squares it, and then subtracts 4 . a) Write \(h\) and \(k\) as equations. b) Graph \(h\) and \(k\) on the same axes. c) Are \(h\) and \(k\) the same function?

Graph. $$ f(x)=\left\\{\begin{array}{ll} 6, & \text { for } x=-2 \\ x^{2}, & \text { for } x \neq-2 \end{array}\right. $$

Rewrite each of the following as an equivalent expression with rational exponents. $$ \sqrt{x^{5}} $$

Refer to Example \(14 .\) The home range, in hectares, of an omnivorous mammal of mass \(w\) grams is given by $$H(w)=0.059 w^{0.92}$$ (Source: Harestad, A. S., and Bunnel, F. L., "Home Range and Body Weight-A Reevaluation," Ecology, Vol. 60, No. 2 (April, 1979), pp. 405-418.) Complete the table of approximate function values and graph the function. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline w & 0 & 1000 & 2000 & 3000 & 4000 & 5000 & 6000 & 7000 \\ \hline H(w) & 0 & 34.0 & & & & & & \\ \hline \end{array} $$

Graph. $$ g(x)=\left\\{\begin{array}{ll} \frac{1}{2} x-1, & \text { for } x<2 \\ -4, & \text { for } x=2 \\ x-3, & \text { for } x>2 \end{array}\right. $$
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