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Chapter 12: Problem 80

Graph by using the slope and \(y\) -intercept. (GRAPH CAN'T COPY) $$3 x-y=1$$

Short Answer

Expert verified
The slope of the line is 3 and the y-intercept is -1. The line can be graphed by first plotting the y-intercept and then using the slope to draw the line.

Step by step solution

01

Converting to Slope-Intercept Form

First, the given equation \(3x - y = 1\) should be rearranged into the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To do this, subtract \(3x\) from both sides of the equation to get \(-y = -3x + 1\). Then, multiply everything by \(-1\) to get \(y = 3x - 1\). So, the slope \(m\) is 3 and the y-intercept \(b\) is -1.
02

Plotting the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. From step 1 we know that this is the point (0, -1), so plot this point on the graph.
03

Using the Slope to Draw the Line

The slope is the rise over the run, so from the y-intercept move 3 units up (rise) and 1 unit to the right (run). Plot this new point. Then draw a line that passes through both 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.

Slope-Intercept Form
The slope-intercept form of a linear equation is a handy way to express straight-line equations. It is written as:
  • \( y = mx + b \)
Here, \(m\) represents the slope, which tells us how steep the line is. The \(b\) term is the y-intercept, showing where the line crosses the y-axis. This form lets you quickly identify these critical attributes of the line.
To use the slope-intercept form effectively, start by rearranging the given equation so that \(y\) stands alone on one side of the equation. This transformation is crucial because it allows you to directly read the slope and y-intercept from the equation once it's in the right format.
Y-Intercept
The y-intercept is a vital starting point when you're graphing linear equations. It is the value of \(y\) when \(x = 0\). In other words, it tells you where the line intersects the y-axis.
In our equation that has been rearranged into slope-intercept form \(y = 3x - 1\), the y-intercept is \(-1\). This means that the line will intersect the y-axis at the point (0, -1).
  • Always plot this point on the graph first.
It anchors the line on the graph and provides an essential reference point for plotting additional points using the slope. Without the y-intercept as a guide, it would be like trying to find a location on a map without a starting point.
Plotting Points
Plotting points accurately is vital to graphing linear equations effectively. After identifying the y-intercept, you use the slope to find another point on the line.
The slope is expressed as 'rise over run' (\( m = \frac{\text{rise}}{\text{run}} \)). For the slope \(3\), this means you rise (go up) 3 units and run (move right) 1 unit from the y-intercept. This gives you a second point on the graph.
  • For example, from (0, -1), moving up 3 units takes you to \(y = 2\), and moving right 1 unit brings you to \(x = 1\), giving you the point (1, 2).
  • Plot this second point.
Once both these points are plotted, you can draw a line through them. This line represents the set of all solutions to the equation.
Equation Rearrangement
Rearranging an equation into the slope-intercept form requires careful manipulation. The goal is to isolate \(y\) on one side of the equation.
Starting with the equation \(3x - y = 1\), subtract \(3x\) from both sides to separate variables and coefficients:
  • \(-y = -3x + 1\).
Next, multiply through by \(-1\) to make \(y\) positive:
  • \(y = 3x - 1\).
This places the equation neatly into the slope-intercept form, \(y = mx + b\), allowing easy reading of the slope and y-intercept.
Rearranging equations may seem complex at first, but it becomes straightforward with practice. Always perform the same operations on both sides of the equation to maintain balance, and check your work to ensure accuracy.

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Most popular questions from this chapter

Evaluate a function. Given \(f(x)=\frac{x}{x+5},\) find \(f(-3)\).

Use the function \(f(x)=x^{2}-4 .\) For the given condition on \(a\) determine whether \(f(a)\) must be positive, must be negative, or could be either positive or negative. $$a>2$$

Determine whether the line through \(P_{1}\) and \(P_{2}\) is parallel, perpendicular, or neither parallel nor perpendicular to the line through \(Q_{1}\) and \(Q_{2}\). $$P_{1}(7,-1), P_{2}(-4,6) ; Q_{1}(3,0), Q_{2}(-5,3)$$

Use the function \(f(x)=x^{2}-4 .\) For the given condition on \(a\) determine whether \(f(a)\) must be positive, must be negative, or could be either positive or negative. $$a<-2$$

Graph. (GRAPH CANNOT COPY) $$x=3$$
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