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Chapter 5: Problem 40

Identify the slope and \(y\) -intercept, then graph the line. $$y=-2 x+7$$

Short Answer

Expert verified
The slope (m) of the given equation \(y = -2x + 7\) is -2 and the y-intercept (b) is 7. Plot the y-intercept at the point (0,7) and use the slope \(\frac{-2}{1}\) (down 2 units and 1 unit to the right) to draw the line.

Step by step solution

01

Identify the slope and y-intercept

The given equation is already in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. In the equation \(y = -2x + 7\), the slope (m) is -2 and the y-intercept (b) is 7.
02

Plot the y-intercept

Now that we have identified the y-intercept as 7, we can plot this point on a graph. The y-intercept is the point where the line crosses the y-axis, so this point is at (0,7).
03

Use the slope to draw the line

With the point (0,7) plotted on the graph, we can now use the slope to draw the line. The slope, -2, can be expressed as a fraction: \(\frac{-2}{1}\). This means that the line has a rise of -2 (goes down 2 units) and a run of 1 (goes right 1 unit). Starting at the y-intercept (0,7), move down 2 units and 1 unit to the right. This will take you to the point (1,5). Plot this point on the graph. You can continue using the slope to find more points on the line if needed.
04

Draw the line

Now that you have multiple points plotted on the graph, connect them to draw the line. Make sure the line passes through the y-intercept and follows the slope by having a consistent rise of -2 and a run of 1.

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Key Concepts

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

Graphing Linear Equations
Let's explore graphing linear equations, a fundamental concept in algebra. A linear equation, like the one given as \( y = -2x + 7 \), describes a straight line on the graph. The general form for a linear equation in the context of graphing is the slope-intercept form: \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept.

To graph a linear equation:
  • Identify the y-intercept, which is the point where the line crosses the y-axis.
  • Use the slope to determine how the line moves across the graph.
  • Plot the y-intercept and use the slope to find additional points on the line.
Connecting these points will give you the straight line representation of the equation.In this exercise, the line can be formed by starting at the y-intercept (0,7) and using the slope to guide further plotting.
Slope of a Line
The slope of a line is a measure of its steepness and direction. In the equation \( y = -2x + 7 \), the slope \( m \) is -2. This tells us how fast and in what direction the line moves.

The slope is represented as a fraction, \( \frac{rise}{run} \), where:
  • Rise: Indicates how many units the line moves up or down.
  • Run: Shows how many units the line moves left or right.
For this line, the slope \( -2 \) can be rewritten as \( \frac{-2}{1} \). This means starting from any point on the line, you move down 2 units (because it's negative), and right 1 unit to arrive at another point on the line.
Understanding the slope helps to quickly chart the direction and angle of the line.
Y-Intercept
The y-intercept is a crucial part of graphing, as it provides a starting point for plotting the line. In the equation \( y = -2x + 7 \), the y-intercept is the constant term, which is 7.

This means the line will cross the y-axis at the point (0,7). This is always where the \( x \) value is zero.
Plotting the y-intercept:
  • Start at the origin (0,0) on a graph.
  • Move vertically to where \( y = 7 \), giving you the point (0,7).
From this point, the slope helps to plot the rest of the line, ensuring that every point follows the pattern set by the slope. This starting point is essential for accurately drawing the straight line dictated by the equation.

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Most popular questions from this chapter

Solve each system \(\begin{aligned} 7 x+8 y-z &=16 \\\\-\frac{1}{2} x-2 y+\frac{3}{2} z &=1 \\\ \frac{4}{3} x+4 y-3 z &=-\frac{2}{3} \end{aligned}\)

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slopeintercept form or in standard form, as indicated. $$-2 x+3 y=15 ;(-5,7) ; \text { standard form }$$

The measure of the smallest angle of a triangle is onethird the measure of the largest angle. The middle angle measures \(30^{\circ}\) less than the largest angle. Find the measures of the angles of the triangle. (Hint: Recall that the sum of the measures of the angles of a triangle is \(180^{\circ} .\) )

Write a system of equations and solve. Six White Castle hamburgers and one small order of french fries contain 955 calories. Eight hamburgers and two orders of fries contain \(1350 .\) Determine how many calories are in a White Castle hamburger and in a small order of french fries. (Source: www.whitecastle com)

Solve each system \(\begin{aligned} 2 x+3 y-z &=0 \\ x-4 y-2 z &=-5 \\\\-4 x+5 y+3 z &=-4 \end{aligned}\)
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