Elementary Algebra
6.3 Graphing Lines
To graph a line by points, solve the equation for y, then find three solution points. Two points are really only needed to graph a line, but the third point is used for accuracy in case a mathematical error is made.
| How to Graph a Line by Points. |
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| 1. Solve the equation for y. |
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| 2. Find three solution points. |
| 3. Plot the three points on the graph and draw the line. |
Example 1:
Graph y = 2x + 1
Solution:
y = 2x + 1
Since the problem is already solved for y, find three solution points.
Use the equation to find where the line is placed on the graph. The equation is a formula that knows the exact location of each point.
Choose three x values, insert them into the equation one at a time and find the corresponding y values by solving the equation for y.
The easiest numbers to choose are x = - 1, x = 0, and x = 1.
Any x values may be selected. These three values just happen to create the least amount of work, and usually the points will fall within a small grid.
Find the corresponding y values for each of the values x = - 1, x = 0, and x = 1.
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y = 2x + 1 |
y = 2x + 1 |
y = 2x + 1 |
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Let x = - 1. |
Let x = 0. |
Let x = 1. |
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y=2(-1) + 1 |
y = 2(0) + 1 |
y = 2(1) + 1 |
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y = - 2 + 1 |
y = 0 + 1 |
y = 2 + 1 |
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y = - 1 |
y = 1 |
y = 3 |
Therefore the following points on the line are obtained.
Plot the points (-1,-1), (0,1), (1,3). All the points must fall along a line. If they do not, then a mathematical error has occurred and you must recalculate your points.
Take note that an x value of 10 can be chosen, but the corresponding y value is 21. The point (10, 21) is a point on the line, but the point is too large to plot on our small graph.
Graph: y = - 3x - 2
Thus the x values of x = - 1, x = 0 and x = 1 produced points that were easy to plot on our small graph.
Student #1:
Graph the line, and then click Answer to view the solution
Answer
Example 2:
Graph: 
Solution:
When the coefficient of the x term is a fraction, the numbers x = - 1, x = 0, and x = 1 are no longer the easiest points to plot. They still produce points along the line, but the points will be fractions. To obtain points that are not fractions, choose the denominator of the fraction, its negative, and zero.
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Let x = - 3. |
Let x = 0. |
Let x = 3. |
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y = 2 - 4 |
y = 0 - 4 |
y = -2 - 4 |
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y = - 2 |
y = - 4 |
y = - 6 |
Therefore the following points on the line are obtained.
Plot the points (-3, -2), (0, -4), (3, -6), and draw the line.
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Graphing a Line by the Slope Intercept Method
| The Slope Intercept Form. |
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| The slope intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept. |
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Example 3:
Find the slope and the y-intercept of the line y = 4x - 13.
Solution:
Use the form y = mx + b to find the slope and y-intercept.
y = mx + b
y = 4x - 13
The slope is 4 and the y-intercept is - 13.
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Example 4:
Find the slope and the y-intercept of the line 2x - 3y = 12.
Solution:
2x - 3y = 12
Solve for y.
- 3y = -2x + 12
The slope is 2/3 and the y-intercept is - 4.
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Consider the last equation:
The slope is 2/3 and the y-intercept is - 4. We can graph the line using the idea of y = mx + b.
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How to Graph a Line using the Slope Intercept Method
y = mx + b |
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| 1. Solve the equation for y, and determine the slope and y-intercept. |
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2. Plot the y-intercept b on the y-axis.
(Make sure you plot b on the y-axis and not the x-axis!) |
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3. From the y-intercept b, find another point on the line using the
rise over the run technique. |
| 4. Draw the line connecting the two points. |
Example 5:
Graph: 
Step 1:
The equation is already solved for y.Step 2:
Plot - 4 on the y-axis. Make sure - 4 is plotted on the y-axis and not the x-axis. This is one of students most common mistakes.
Step 3:
From the y-intercept - 4, find another point on the line using the rise over the run technique. The rise is 2, and the run is 3. Rise up 2 and then run to the right 3 units. More points can be determined by another rise of 2, and run of 3.
Step 4:
Draw the line to connect the points.
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Example 6:
Graph: 
Solution:
Step 1:
The equation is already solved for y.Step 2:
Plot the y-intercept + 2 on the y-axis.
Step 4:
From the 2 on the y-axis, rise 1 and run 3 to the right.
Step 5:
Draw the line that connects the two points.
Graph: 
Graph the line and then click Answer to view the solution.
Answer
Graph: 
Graph the line and then click Answer to view the solution.
Answer
Graph: 4x + 5y = 15
Graph the line and then click Answer to view the solution.
Answer
Graph: y = 2x - 3
Graph the line and then click Answer to view the solution.
Answer
Solution:
y = - 3x - 2
Plot points:
Find the corresponding y values for each of the x = - 1, x = 0, and x = 1.
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y = - 3x - 2 |
y = - 3x - 2 |
y = - 3x - 2 |
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Let x = - 1. |
Let x = 0. |
Let x = 1. |
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y=-3(-1) - 2 |
y = -3(0) - 2 |
y = -3(1) - 2 |
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y = 3 - 2 |
y = 0 - 2 |
y = -3 - 2 |
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y = 1 |
y = -2 |
y = -5 |
Therefore the following points on the line are obtained.
Plot the points (-1, 1), (0, -2), (1, -5), and draw the line.
Go Back to Lesson
Solution:
Don’t forget to plot +1 on the Y-AXIS, and not the x-axis!
Go Back to Lesson
Solution:
Don’t forget to plot -2 on the Y-AXIS, and not the x-axis!
Go Back to Lesson
Solution:
4x + 5y = 15
Step 1:
Solve for y, and determine the slope and y-intercept.5y = - 4x + 15
Step 2:
Plot 3 on the y-axis.Don’t forget to plot +3 on the Y-AXIS, and not the x-axis!
Step 3 and 4:
Rise down 4 units, and then run to the right 5 units.
Go Back to Lesson
Solution:
y = 2x - 3
The slope is 2 or (2/1), and the y-intercept is -3.
Go Back to Lesson
