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Chapter 2: Problem 96

Find the slope and y-intercept of each line. Graph the line. $$ x+y=0 $$

Short Answer

Expert verified
Slope: -1, y-intercept: 0.

Step by step solution

01

Rewrite the equation in slope-intercept form

The slope-intercept form of an equation is given by \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. Starting with the equation \(x + y = 0\), we need to isolate \(y\) on one side of the equation.
02

Isolate y

To isolate \(y\), subtract \(x\) from both sides: \(y = -x\). Now the equation is in slope-intercept form.
03

Identify the slope and y-intercept

In the equation \(y = -x\), the coefficient of \(x\) is \(-1\). Therefore, the slope \(m = -1\). There is no constant term, so the y-intercept \(b = 0\).
04

Graph the line

To graph the line, start at the y-intercept (0, 0). From there, use the slope \(-1\), which means for every 1 unit you move to the right, move 1 unit down. Plot additional points and draw a straight line through these points.

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Key Concepts

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

slope-intercept form
Understanding the slope-intercept form is essential for solving linear equations. It is typically written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept. Starting with an equation, like the given \(x + y = 0\), our goal is to rewrite it so it matches this form. By isolating \(y\) (getting it alone on one side of the equation), we can see the values of \(m\) and \(b\). This makes it easy to identify key features of the line.
y-intercept
The y-intercept is where the line crosses the y-axis. In the slope-intercept form, the y-intercept is \(b\). For our equation \(y = -x\), there's no constant term added to \(x\), making \(b = 0\). This means the line passes through the origin, or the point \( (0, 0) \). Always start graphing the line from the y-intercept, as it gives an anchor point for plotting further points along the slope.
graphing lines
Graphing a line involves plotting it on a coordinate plane. Start at the y-intercept \( (0, 0) \) for \(y = -x\). With a slope \(m = -1\), move 1 unit to the right and 1 unit down for the next point (following the negative slope). Connect these dots to draw the line. The process highlights how understanding slope and y-intercept aids in visualizing and accurately drawing linear equations. Consistently check that the points align correctly along the slope to ensure accuracy.

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Most popular questions from this chapter

The equations of two lines are given. Determine whether the lines are parallel, perpendicular, or neither. $$ \begin{array}{l} y=4 x+5 \\ y=-4 x+2 \end{array} $$

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ 0.2 x-0.5 y=1 $$

Solve each equation. \(6 x-2(x-4)=24\)

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