8.5: Graph Linear Equations in Two Variables
By the end of this section, you will be able to:
- Plot points in a rectangular coordinate system
- Graph a linear equation by plotting points
- Graph vertical and horizontal lines
- Find the \(x\)- and \(y\)-intercepts
- Graph a line using the intercepts
Graph a Linear Equation by Plotting Points
There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.
Graph the equation \(y=2x+1\) by plotting points.
Solution:
Graph the equation by plotting points: \(y=2x−3\).
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Graph the equation by plotting points: \(y=−2x+4\).
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The steps to take when graphing a linear equation by plotting points are summarized here.
- Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.
It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.
If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between these illustrations.
When an equation includes a fraction as the coefficient of \(x,\) we can still substitute any numbers for \(x.\) But the arithmetic is easier if we make “good” choices for the values of \(x.\) This way we will avoid fractional answers, which are hard to graph precisely.
Graph the equation: \(y=\frac{1}{2}x+3\).
Solution:
Find three points that are solutions to the equation. Since this equation has the fraction \(\dfrac{1}{2}\) as a coefficient of \(x,\) we will choose values of \(x\) carefully. We will use zero as one choice and multiples of \(2\) for the other choices. Why are multiples of two a good choice for values of \(x\)? By choosing multiples of \(2\) the multiplication by \(\dfrac{1}{2}\) simplifies to a whole number
The points are shown in Table .
| \(y=\frac{1}{2}x+3\) | ||
| \(x\) | \(y\) | \((x,y)\) |
| 0 | 3 | \((0,3)\) |
| 2 | 4 | \((2,4)\) |
| 4 | 5 | \((4,5)\) |
Plot the points, check that they line up, and draw the line.
Graph the equation: \(y=\frac{1}{3}x−1\).
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Graph the equation: \(y=\frac{1}{4}x+2\).
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Graph Vertical and Horizontal Lines
Some linear equations have only one variable. They may have just \(x\) and no \(y,\) or just \(y\) without an \(x.\) This changes how we make a table of values to get the points to plot.
Let’s consider the equation \(x=−3\). This equation has only one variable, \(x.\) The equation says that \(x\)is always equal to \(−3\), so its value does not depend on \(y.\) No matter what is the value of \(y,\) the value of \(x\) is always \(−3\).
So to make a table of values, write \(−3\) in for all the \(x\)-values. Then choose any values for \(y.\) Since \(x\) does not depend on \(y,\) you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the \(y\)-coordinates. See Table.
| \(x=−3\) | ||
| \(x\) | \(y\) | \((x,y)\) |
| \(−3\) | 1 | \((−3,1)\) |
| \(−3\) | 2 | \((−3,2)\) |
| \((−3,)\) | 3 | \((−3,3)\) |
Plot the points from the table and connect them with a straight line. Notice that we have graphed a vertical line .
What if the equation has \(y\) but no \(x\)? Let’s graph the equation \(y=4\). This time the y- value is a constant, so in this equation, \(y\) does not depend on \(x.\) Fill in \(4\) for all the \(y\)’s in Table and then choose any values for \(x.\) We’ll use 0, 2, and 4 for the \(x\)-coordinates.
| \(y=4\) | ||
| \(x\) | \(y\) | \((x,y)\) |
| 0 | 4 | \((0,4)\) |
| 2 | 4 | \((2,4)\) |
| 4 | 4 | \((4,4)\) |
In this figure, we have graphed a horizontal line passing through the \(y\)-axis at \(4.\)
A vertical line is the graph of an equation of the form \(x=a\).
The line passes through the \(x\)-axis at \((a,0)\).
A horizontal line is the graph of an equation of the form \(y=b\).
The line passes through the \(y\)-axis at \((0,b)\).
Graph: a. \(x=2\) b. \(y=−1\).
Solution
a. The equation has only one variable, \(x,\) and \(x\) is always equal to \(2.\) We create a table where \(x\) is always \(2\) and then put in any values for \(y.\) The graph is a vertical line passing through the \(x\)-axis at \(2.\)
| \(x\) | \(y\) | \((x,y)\) |
|---|---|---|
| 2 | 1 | \((2,1)\) |
| 2 | 2 | \((2,2)\) |
| 2 | 3 | \((2,3)\) |
b. Similarly, the equation \(y=−1\) has only one variable, \(y\). The value of \(y\) is constant. All the ordered pairs in the next table have the same \(y\)-coordinate. The graph is a horizontal line passing through the \(y\)-axis at \(−1.\)
| \(\mathbf{x}\) | \(\mathbf{ y}\) | \(\mathbf{(x,y)}\) |
|---|---|---|
| 0 | \(−1\) | \((0,−1)\) |
| 3 | \(−1\) | \((3,−1)\) |
| \(−3\) | \(−1\) | \((−3,−1)\) |
G raph the equations: a. \(x=5\) b. \(y=−4\).
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a.
b.
G raph the equations: a. \(x=−2\) b. \(y=3\).
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a.
b.
What is the difference between the equations \(y=4x\) and \(y=4\)?
The equation \(y=4x\) has both \(x\) and \(y.\) The value of \(y\) depends on the value of \(x,\) so the \(y\)-coordinate changes according to the value of \(x.\) The equation \(y=4\) has only one variable. The value of \(y\) is constant, it does not depend on the value of \(x,\) so the \(y\)-coordinate is always \(4.\)
Notice, in the graph, the equation \(y=4x\) gives a slanted line, while \(y=4\) gives a horizontal line.
Graph \(y=−3x\) and \(y=−3\) in the same rectangular coordinate system.
Solution:
We notice that the first equation has the variable \(x,\) while the second does not. We make a table of points for each equation and then graph the lines. The two graphs are shown.
Graph the equations in the same rectangular coordinate system: \(y=−4x\) and \(y=−4\).
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Graph the equations in the same rectangular coordinate system: \(y=3\)and \(y=3x\).
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Find \(x\)- and \(y\)-intercepts
Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.
At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the \(x\)-axis and the \(y\)-axis. These points are called the intercepts of a line .
The points where a line crosses the \(x\)-axis and the \(y\)-axis are called the intercepts of the line .
Let’s look at the graphs of the lines.
First, notice where each of these lines crosses the \(x\)-axis. See Table .
Now, let’s look at the points where these lines cross the \(y\)-axis.
| Figure |
The line crosses
the \(x\)-axis at: |
Ordered pair
for this point |
The line crosses
the y- axis at: |
Ordered pair
for this point |
|---|---|---|---|---|
| Figure (a) | \(3\) | \((3,0)\) | \(6\) | \((0,6)\) |
| Figure (b) | \(4\) | \((4,0)\) | \(−3\) | \((0,−3)\) |
| Figure (c) | \(5\) | \((5,0)\) | \(−5\) | \((0,5)\) |
| Figure (d) | \(0\) | \((0,0)\) | \(0\) | \((0,0)\) |
| General Figure | \(a\) | \((a,0)\) | \(b\) | \((0,b)\) |
Do you see a pattern?
For each line, the \(y\)-coordinate of the point where the line crosses the \(x\)-axis is zero. The point where the line crosses the \(x\)-axis has the form \((a,0)\) and is called the \(x\) -intercept of the line. The \(x\)-intercept occurs when \(y\) is zero.
In each line, the \(x\) - coordinate of the point where the line crosses the \(y\)-axis is zero. The point where the line crosses the \(y\)-axis has the form \((0,b)\) and is called the \(y\) -intercept of the line. The \(y\)-intercept occurs when \(x\) is zero.
The \(x\)-intercept is the point \((a,0)\) where the line crosses the \(x\)-axis.
The \(y\)-intercept is the point \((0,b)\) where the line crosses the \(y\)-axis.
Find the \(x\)- and \(y\)-intercepts on each graph shown.
Solution:
a. The graph crosses the \(x\)-axis at the point \((4,0)\). The
x-
intercept is \((4,0)\).
The graph crosses the \(y\)-axis at the point \((0,2)\). The \(y\)-intercept is \((0,2)\).
b. The graph crosses the \(x\)-axis at the point \((2,0)\). The \(x\)-intercept is \((2,0)\).
The graph crosses the \(y\)-axis at the point \((0,−6)\). The \(y\)-intercept is \((0,−6)\).
c. The graph crosses the \(x\)-axis at the point \((−5,0)\). The \(x\)-intercept is \((−5,0)\).
The graph crosses the \(y\)-axis at the point \((0,−5)\). The \(y\)-intercept is \((0,−5)\).
Find the \(x\)- and \(y\)-intercepts on the graph.
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\(x\)-intercept: \((2,0)\),
\(y\)-intercept: \((0,−2)\)
Find the \(x\)- and \(y\)-intercepts on the graph.
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\(x\)-intercept: \((3,0)\),
\(y\)-intercept: \((0,2)\)
Recognizing that the \(x\) -intercept occurs when \(y\) is zero and that the \(y\) -intercept occurs when \(x\) is zero, gives us a method to find the intercepts of a line from its equation. To find the \(x\) -intercept, let \(y=0\) and solve for \(x.\) To find the \(y\) -intercept, let \(x=0\) and solve for \(y.\)
Use the equation of the line. To find:
- the \(x\)-intercept of the line, let \(y=0\) and solve for \(x\).
- the \(y\)-intercept of the line, let \(x=0\) and solve for \(y\).
Find the intercepts of \(2x+y=8\).
Solution:
We will let \(y=0\) to find the \(x\)-intercept, and let \(x=0\) to find the \(y\)-intercept. We will fill in a table, which reminds us of what we need to find.
| To find the \(x\)-intercept, let \(y=0\). | |
| \(2x+y=8\) | |
| Let \(y=0\). | \(2x+{\color{red}0}=8\) |
| Simplify. | \(2x=8\) |
| \(x=4\) | |
| The \(x\)-intercept is: | \((4,0)\) |
| To find the \(y\)-intercept, let \(x=0\). | |
| \(2x+y=8\) | |
| Let \(x=0\). | \(2 ( {\color{red}0}) + y = 8\) |
| Simplify. | \(0 + y = 8\) |
| \(y=8\) | |
| The \(y\)-intercept is: | \((0,8)\) |
The intercepts are the points \((4,0)\) and \((0,8)\) as shown in the table.
| \(2x+y=8\) | |
| \(x\) | \(y\) |
| 4 | 0 |
| 0 | 8 |
Find the intercepts: \(3x+y=12\).
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\(x\)-intercept: \((4,0)\),
\(y\)-intercept: \((0,12)\)
Find the intercepts: \(x+4y=8\).
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\(x\)-intercept: \((8,0)\),
\(y\)-intercept: \((0,2)\)
Graph a Line Using the Intercepts
To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.
Graph \(–x+2y=6\) using the intercepts.
Solution:
Graph using the intercepts: \(x–2y=4\).
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Graph using the intercepts: \(–x+3y=6\).
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The steps to graph a linear equation using the intercepts are summarized here.
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Find the \(x\)- and \(y\)-intercepts of the line.
- Let y=0y=0 and solve for \(x\).
- Let x=0x=0 and solve for \(y\).
- Find a third solution to the equation.
- Plot the three points and check that they line up.
- Draw the line.
Graph \(4x−3y=12\) using the intercepts.
Solution:
Find the intercepts and a third point.
We list the points in the table and show the graph.
| \(4x−3y=12\) | ||
| \(x\) | \(y\) | \((x,y)\) |
| 3 | 0 | \((3,0)\) |
| 0 | \(−4\) | \((0,−4)\) |
| 6 | 4 | \((6,4)\) |
Graph using the intercepts: \(5x−2y=10\).
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Graph using the intercepts: \(3x−4y=12\).
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When the line passes through the origin, the \(x\)-intercept and the \(y\)-intercept are the same point.
Graph \(y=5x\) using the intercepts.
Solution:
This line has only one intercept. It is the point \((0,0)\).
To ensure accuracy, we need to plot three points. Since the \(x\)- and \(y\)-intercepts are the same point, we need two more points to graph the line.
The resulting three points are summarized in the table.
| \(y=5x\) | ||
| \(x\) | \(y\) | \((x,y)\) |
| 0 | 0 | \((0,0)\) |
| 1 | 5 | \((1,5)\) |
| \(−1\) | \(−5\) | \((−1,−5)\) |
Plot the three points, check that they line up, and draw the line.
Graph using the intercepts: \(y=4x\).
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Graph the intercepts: \(y=−x\).
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Key Concepts
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Points on the Axes
- Points with a \(y\)-coordinate equal to \(0\) are on the \(x\)-axis, and have coordinates \((a,0)\).
- Points with an \(x\)-coordinate equal to \(0\) are on the \(y\)-axis, and have coordinates \((0,b)\).
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Quadrant
Quadrant I Quadrant II Quadrant III Quadrant IV \((x,y)\) \((x,y)\) \((x,y)\) \((x,y)\) \((+,+)\) \((-,+)\) \((-,-)\) \((+,-)\) -
Graph of a Linear Equation:
The graph of a linear equation \(Ax+By=C\) is a straight line.
Every point on the line is a solution of the equation.
Every solution of this equation is a point on this line. -
How to graph a linear equation by plotting points.
- Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.
-
\(x\)-intercept and \(y\)-intercept of a Line
- The \(x\)-intercept is the point \((a,0)\) where the line crosses the \(x\)-axis.
- The \(y\)-intercept is the point \((0,b)\) where the line crosses the \(y\)-axis.
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Find the \(x\)- and \(y\)-intercepts from the Equation of a Line
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Use the equation of the line. To find:
the \(x\)-intercept of the line, let \(y=0\) and solve for \(x.\)
the \(y\)-intercept of the line, let \(x=0\) and solve for \(y.\)
-
Use the equation of the line. To find:
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How to graph a linear equation using the intercepts.
-
Find the \(x\)- and \(y\)-intercepts of the line.
Let \(y=0\) and solve for \(x.\)
Let \(x=0\) and solve for \(y.\) - Find a third solution to the equation.
- Plot the three points and check that they line up.
- Draw the line.
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Find the \(x\)- and \(y\)-intercepts of the line.
Glossary
- horizontal line
- A horizontal line is the graph of an equation of the form \(y=b.\) The line passes through the \(y\)-axis at \((0,b).\)
- intercepts of a line
- The points where a line crosses the \(x\)-axis and the \(y\)-axis are called the intercepts of the line.
- linear equation
- An equation of the form \(Ax+By=C,\) where \(A\) and \(B\) are not both zero, is called a linear equation in two variables.
- ordered pair
- An ordered pair, \((x,y),\) gives the coordinates of a point in a rectangular coordinate system. The first number is the \(x\)-coordinate. The second number is the \(y\)-coordinate.
- origin
- The point \((0,0)\) is called the origin. It is the point where the \(x\)-axis and \(y\)-axis intersect.
- solution of a linear equation in two variables
- An ordered pair \((x,y)\) is a solution of the linear equation \(Ax+By=C,\) if the equation is a true statement when the \(x\)- and \(y\)-values of the ordered pair are substituted into the equation.
- standard form of a linear equation
- A linear equation is in standard form when it is written \(Ax+By=C.\)
- vertical line
- A vertical line is the graph of an equation of the form \(x=a.\) The line passes through the \(x\)-axis at \((a,0).\)