/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 60 Graph each equation using any me... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 60

Graph each equation using any method. $$y+x=-2$$

Short Answer

Expert verified
The line crosses the y-axis at y = -2 and drops one unit for each unit moved to the right.

Step by step solution

01

Rewrite the equation into slope-intercept form

Rearrange the given equation \(y + x = -2\) into slope-intercept form (\(y = mx + b\)). To do this, subtract \(x\) from both sides: \(y = -x - 2\). Thus, the slope (\(m\)) is -1, and the \(y\)-intercept (\(b\)) is -2.
02

Plot the y-intercept

On a Cartesian plane, mark the point where \(y = -2\). This is the y-intercept which serves as the starting point for the graphing process.
03

Use the slope to find the next point

The slope in this case is -1, which implies a one-unit decrease in the y-coordinate for a one-unit increase in the x-coordinate. So, move one step right and one step down from the y-intercept to locate the next point.
04

Draw the line

Use the two points found in step 2 and step 3, and draw a straight line through these points. This line is the graph of the given equation.

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Key Concepts

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

Understanding the Slope-Intercept Form
The slope-intercept form is a linear equation written as \(y = mx + b\). This expression is powerful for graphing because it clearly identifies two critical components: the slope \(m\) and the \(y\)-intercept \(b\).
  • The slope \(m\) indicates the steepness or tilt of the line. A positive slope means the line ascends from left to right, while a negative slope means it descends.
  • The \(y\)-intercept \(b\), is the point where the line crosses the \(y\)-axis. It's where \(x = 0\).
By converting any linear equation into this form, you gain all necessary information to graph the equation quickly. For instance, the equation \(y + x = -2\) is rearranged to \(y = -x - 2\), revealing the slope as -1 and the \(y\)-intercept as -2.
Importance of the Y-Intercept
The \(y\)-intercept is a fundamental point that gives a graph its starting position on the vertically oriented \(y\)-axis. In the equation \(y = -x - 2\), \(b = -2\) means this line crosses the \(y\)-axis at -2.
  • This point is crucial because it is commonly the first point you plot when graphing a line.
  • Knowing the \(y\)-intercept helps you understand the line's vertical starting point without needing any calculation.
Once the \(y\)-intercept is established on a graph, you use the slope to determine additional points on the line.
Decoding the Slope of a Line
The slope of a line, represented by \(m\) in the slope-intercept form \(y = mx + b\), describes how the line moves across the Cartesian plane. In essence, it tells how much \(y\) changes for a specific change in \(x\).
  • A slope of -1, as in \(y = -x - 2\), indicates that for every unit increase in \(x\), the value of \(y\) decreases by one unit.
  • The slope can have three major effects: tilting the line upwards, downwards, or keeping it flat (like in horizontal lines \(m = 0\)).
A negative slope like -1 will result in a downward slanting line as you move from left to right across the graph.
Navigating the Cartesian Plane
The Cartesian plane is a two-dimensional surface defined by a horizontal \(x\)-axis and a vertical \(y\)-axis. It鈥檚 the playground where all graphing happens. Each point on the plane is pinpointed by a coordinate \((x, y)\).
  • Positive values on the \(x\)-axis go to the right, and positive values on the \(y\)-axis go up.
  • Negative values on the \(x\)-axis go to the left, and negative values on the \(y\)-axis go down.
To graph an equation like \(y = -x - 2\), you'd start at the \(y\)-intercept point on the \(y\)-axis, then use the slope to find other points, marking the pathway of the graph through this coordinate system. This setup makes it easy to visualize relationships between variables in equations.

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Most popular questions from this chapter

Can inverse variation be defined as \(x y=k,\) rather than \(y=\frac{k}{x} ?\)

Set up a variation equation and solve for the requested value. The distance traveled by an object in free fall varies directly with the square of the time that it falls. If the object falls 256 feet in 4 seconds, how far will it fall in 6 seconds?

Physiology Physiologists have found that a woman's height \(h\) in inches can be approximated using the linear equation \(h=3.9 r+28.9,\) where \(r\) represents the length of her radius bone in inches. a. Complete the table of values (round to the nearest tenth), and then graph the equation. b. Complete this sentence: From the graph, we see that the longer the radius bone, the. c. From the graph, estimate the height of a girl whose radius bone is 10 inches long.

Fill in the blanks. Assume that \(k\) is a constant. The equation \(y=k x\) represents _____ variation.

Express each combined variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 4. (OBJECTIVE 4) \(p\) varies directly with \(q\) and inversely with \(r .\) If \(p=5\) when \(q=1\) and \(r=6,\) find \(p\) when \(q=5\) and \(r=10\)
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