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Chapter 2: Problem 31

Give the slope and y-intercept of each line, and graph it. $$4 x-y=7$$

Short Answer

Expert verified
Slope = 4, y-intercept = -7.

Step by step solution

01

Write the equation in slope-intercept form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.Start by rearranging the equation \(4x - y = 7\) to solve for \(y\).
02

Solve for y

Subtract \(4x\) from both sides to isolate \(y\):\[ -y = -4x + 7 \]Next, multiply both sides by -1 to solve for \(y\):\[ y = 4x - 7 \]Now, the equation is in the slope-intercept form \(y = mx + b\).
03

Identify the slope and y-intercept

In the equation \( y = 4x - 7 \), the coefficient of \(x\) gives the slope (\(m\)), and the constant term gives the y-intercept (\(b\)).So, the slope is \( m = 4 \), and the y-intercept is \( b = -7 \).
04

Graph the equation

To graph the equation, start by plotting the y-intercept (0, -7) on the coordinate plane. From this point, use the slope to find another point on the line. The slope \(m = 4\) can be interpreted as 'rise over run,' meaning rise 4 units up and 1 unit to the right. Plot this second point (1, -3). Draw a line through the two points to complete the graph.

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Key Concepts

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

graphing linear equations
Understanding how to graph linear equations is fundamental in algebra. A linear equation is simply an equation that forms a straight line when graphed on a coordinate plane. The standard way to graph these equations is by using the slope-intercept form, which is written as \( y = mx + b \).
Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept.
To graph a linear equation:
  • First, identify the y-intercept and plot it on the y-axis.
  • Then, use the slope to determine the direction of the line.
  • Draw a straight line through these points.
This method helps to accurately represent the linear relationship depicted by the equation. So, in the equation \( 4x - y = 7 \), once it is rearranged into slope-intercept form \( y = 4x - 7 \), we can easily graph it starting from the y-intercept at (0, -7) and using the slope.
finding slope
The slope of a line indicates its steepness and direction. It is represented by the letter \( m \) in the slope-intercept form \( y = mx + b \). The slope is calculated as the 'rise over run', which means the change in the y-coordinates divided by the change in the x-coordinates between two points on the line.
For instance, in our example equation \( y = 4x - 7 \), the slope \( m = 4 \) means that for every unit increase in x, the y value increases by 4 units.
This can be visualized as:
  • 'Rise' 4 units up the y-axis.
  • 'Run' 1 unit to the right on the x-axis.
Hence, from the y-intercept (0, -7), moving up 4 units and right 1 unit gives the next point at (1, -3). Understanding the slope's meaning helps students draw the line accurately on the graph.
y-intercept
The y-intercept is where the line crosses the y-axis. This point is crucial because it gives a fixed starting point on the graph. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by the constant term \( b \).
For our example, the equation \( y = 4x - 7 \) shows that the y-intercept \( b \) is -7. This means the line crosses the y-axis at the point (0, -7).
To summarize:
  • Locate the y-intercept, plot it on the coordinate plane exactly on the y-axis.
  • Use the slope from this point to find another point on the line.
  • Then, draw a line through these points.
Being able to identify the y-intercept helps in easily and correctly graphing the linear equation.

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Most popular questions from this chapter

Use a graphing calculator to solve each linear equation. $$3(2 x+1)-2(x-2)=5$$

Solve each problem. The cost to hire a caterer for a party depends on the number of guests attending. If 100 people attend, the cost per person will be 20 dollars . For each person less than \(100,\) the cost will increase by $$ 5 .\( Assume that no more than 100 people will attend. Let \)x\( represent the number less than 100 who do not attend. For example, if 95 attend, \)x=5\( (a) Write a function defined by \)N(x)\( giving the number of guests. (b) Write a function defined by \)G(x)\( giving the cost per guest. (c) Write a function defined by \)N(x) \cdot G(x)\( for the total cost, \)C(x)$ (d) What is the total cost if 80 people attend?

Give the slope and y-intercept of each line, and graph it. $$y-\frac{3}{2} x-1=0$$

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Write an equation (a) in standard form and (b) in slope-intercept form for the line described through \((-1,4),\) parallel to \(x+3 y=5\)
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