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Chapter 3: Problem 58

Graph each equation using any method. $$y=-2 x+5$$

Short Answer

Expert verified
Start by plotting the y-intercept (0,5). Use the slope (-2/1) to plot a second point (1,3), and then draw a straight line passing through these two points.

Step by step solution

01

Identify the slope and y-intercept

The equation is in the form \[y = mx + b\]. In this case, the slope \(m\) is -2 and the y-intercept \(b\) is 5.
02

Plot the y-intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis, at (0,5). Hence draw a point at this position.
03

Use the slope to find another point

The slope is -2, which can be written as -2/1. This means that for every 1 unit that x increases, y decreases by 2 units. From the y-intercept (0,5), move 1 unit to the right (increasing x) and 2 units down (decreasing y) to find a second point (1,3). Plot this point.
04

Draw the line

Draw a straight line passing through the two points plotted on the graph. This line represents the graph of the equation \(y = -2x + 5\).

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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 is a common way to express the equation of a straight line. It's given by the formula \(y = mx + b\), where:
  • \(m\) represents the slope of the line.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
This format is particularly useful because it directly provides two critical pieces of information needed to graph a linear equation. Understanding \(m\) and \(b\) enables you to quickly sketch any line. In solving the problem, recognizing that \(y = -2x + 5\) is already in slope-intercept form helps determine that the slope is -2 and the y-intercept is 5. This step forms the foundation for graphing the equation.
Plotting Points
Plotting points on a graph is a fundamental skill in graphing linear equations. The y-intercept is always the best starting point because it gives a fixed reference on the y-axis. For example, in the equation \(y = -2x + 5\), the y-intercept is 5, so you start by placing a point at (0,5) on the graph.
Once you have your initial point, the slope tells you how to find more points. Simply follow the rise over run indicated by the slope to plot a second point. These two points will guide you in drawing the entire line accurately. Always ensure your points are plotted precisely for reliability in your graphing results.
Slope Understanding
Understanding the slope is crucial for interpreting how a line behaves. The slope \(m\) of a line describes its steepness and direction:
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope, as in our example with \(m = -2\), implies the line falls as you move rightward.
  • A slope of zero results in a horizontal line, indicating no rise or fall.
  • An undefined slope would be seen with vertical lines.
For the equation \(y = -2x + 5\), the slope is -2, which can be understood as -2/1. This fraction means that for every increase of 1 in \(x\), \(y\) decreases by 2. By moving from the point (0,5) to (1,3), you visualize this decrease, confirming the slope's effect on the line.

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Most popular questions from this chapter

Graph each equation. SEE EXAMPLE 6. (OBJECTIVE 5 ) $$y=4$$

Set up a variation equation and solve for the requested value. The cost of drilling a water well is jointly proportional to the length and diameter of the steel casing. If a 30 -foot well using 4 -inch casing costs \(1,200\)dollar, find the cost of a 35-foot well using 6-inch casing.

Set up a variation equation and solve for the requested value. The force of gravity acting on an object varies directly with the mass of the object. The force on a mass of 5 kilograms is 49 newtons. What is the force acting on a mass of 12 kilograms?

Express each combined variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 4. (OBJECTIVE 4) \(y\) varies directly with the square of \(x\) and inversely with \(z\). If \(y=1\) when \(x=2\) and \(z=10\), find \(y\) when \(x=4\) and \(z=5\).

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. $$P(3,8) \text { and } Q(9,-2)$$
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