Graphical solution of linear systems

In order to solve a linear system using graphs, you must first of all be comfortable with graphing straight lines.

Review of linear graphs

Straight lines are very easy to graph. You just need to know the slope of the line and its y-intercept. Alternatively, you can also graph a straight line if you know both the x and y intercepts (actually, any two points will do, but using the intercepts makes it more convenient).

ExaMple 1

Sketch the graph of the line y=x-2.

Let’s use the intercept method to sketch this graph. To find the y-intercept, set x=0 and obtain y=0-2=-2; to find the x-intercept, set y=0 and solve for x:

    \[0=x-2,\quad x=2\]

Next, we join the y-intercept and the x-intercept with a straight line to obtain the graph of y=x-2 (red line below):

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example 2

Sketch the graph of 2x+3y=6.

Again, let’s use the intercept method to sketch this graph.

  • x-intercept is obtained by putting y=0 and solving for x:

        \[2x+3(0)=6,\quad\implies 2x=6,\quad\implies x=3;\]

  • y-intercept is obtained by putting x=0 and solving for y:

        \[2(0)+3y=6,\quad\implies 3y=6,\quad\implies y=2.\]

We now join the x and y intercepts to obtain the graph below (blue line):

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Example 3

Sketch the graph of 3x-2y-2=0.

Let’s use the slope and the y-intercept this time around. To do so, we first isolate the variable y as follows:

    \[3x-2 =2y,\quad\implies \frac{3x}{2}-\frac{2}{2}=\frac{2y}{2},\quad\implies \frac{3x}{2}-1=y.\]

The equation y=\frac{3x}{2}-1 is now in the form y=mx+b, so its slope is \frac{3}{2} and its y-intercept is -1.

  • Mark the point where y=-1 on the y-axis;
  • From that point (where y=-1), move three units up (which means that you’ll now be at y=2, since -1+3=2);
  • Now move two units to the right (because the slope is 3/2, we have to rise 3 units and run two units to the right. Remember that rise over run thing?)
  • If you did this correctly, the starting coordinate was (0,-1) and you should end up at the point (2,2). Connect these two points together to obtain the graph shown below (green line):

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Linear systems via graphs

With the hope that the above examples on linear graphs suffice, we now move on to solving linear system problems using graphs.

example 4

Use graphs to solve the linear system

(1)   \begin{equation*} x+y=3 \end{equation*}

(2)   \begin{equation*} x-y=1 \end{equation*}

We first graph the two lines and notice where they intersect:

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\boxed{\textrm{As shown in the graph above, the lines intersect at the point (2,1)}}

Example 5

Use graphical method to solve the linear system

(3)   \begin{equation*} 2x+5y=1 \end{equation*}

(4)   \begin{equation*} 6x-4y=3 \end{equation*}

Again, we graph the two lines and notice where they intersect:

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\boxed{\textrm{The lines intersect at (0.5,0), which is the solution}.}
The example above reveals one major limitation of the graphical method, which has to do with determining the point of intersection when such a point does not have integer coordinates. Our example was chosen so that we can easily read the coordinates of the point of intersection. How about when such coordinates are unfamiliar fractions? Like (\frac{3}{7},\frac{5}{13})? Because of this, we do not recommend using the graphical method when the intersection point cannot be easily read from the graph, unless a question specifies so.

Example 6

Determine the point of intersection of the lines

(5)   \begin{equation*} x-y=2 \end{equation*}

(6)   \begin{equation*} 2x-2y=6 \end{equation*}

by sketching their graphs.

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\boxed{\textrm{We see that the lines don't intersect; the system is inconsistent.}}
It appears that an inconsistent system is better understood when visualized by graphs. This is one power of the graphical method.

Example 7

Solve the linear system

(7)   \begin{equation*} x+2y=2 \end{equation*}

(8)   \begin{equation*} 2x+4y-4=0 \end{equation*}

As always, we first graph the two lines:

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Oops \cdots. Only one line is visible — despite the fact that our program graphed two lines (beginning with the one with the red legend and then the one with the green legend). It happened in this case that equations (7) and (8) represent the same line; as such, they are said to be coincident. The number of solutions in this case is infinite.

Example 8

A barnyard has pigs and chickens. Altogether there are 28 legs and 18 eyes. Determine the number of each animal.

Let’s follow the strategy for solving word problems involving linear systems to address this question. Our first step is to introduce variables. So, suppose that the number of pigs is x and that the number of chickens is y.

  • A pig has four legs, while a chicken has two legs.
  • A pig has two eyes, just as a chicken has two eyes.
  • Since there are x pigs and each has four legs, the number of legs from the pigs is 4x.
  • Since there are y chickens and each has two legs, the number of legs contributed by the chickens is 2y.
  • Since the total number of legs is 28, it means that 4x+2y=28.
  • Since the total number of eyes is 18, it means that 2x+2y=18.

The above analysis is probably unnecessary. The whole point is that the question (and the variables we chose) lead to the linear system:

(9)   \begin{equation*} 4x+2y=28\quad\quad\implies 2x+y=14 \end{equation*}

(10)   \begin{equation*} 2x+2y=18\quad\quad\implies x+y=9 \end{equation*}

Because of the relatively big numbers involved, graphing is not recommended for this question. Nevertheless, we’ll go ahead and use graphs being that we’re discussing graphical method.

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From the graph, the point of intersection is (5,4), which corresponds to 5 pigs and 4 chickens.

Example 9

A store sells four ballpoint pens and five pencils for 6.50, while five ballpoint pens and four pencils are sold for 7.00. Find how much a ballpoint pen costs and how much a pencil costs.

We’ve ignored the currency unit. Let the cost of a ballpoint pen be x. Let the cost of a pencil be y. We have the linear system:

(11)   \begin{equation*} 4x+5y=6.50 \end{equation*}

(12)   \begin{equation*} 5x+4y=7.00 \end{equation*}

And then the graphs:

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From the graph, the point of intersection is (x,y)=(1,0.5). So, a ballpoint pen costs 1.00, while a pencil costs 0.50.

Example 10

Find two numbers whose sum is 6 and whose difference is 2.

Let the two numbers be x and y. Without loss of generality, suppose that y is the smaller number. Then:

(13)   \begin{equation*} x+y=6 \end{equation*}

(14)   \begin{equation*} x-y=2 \end{equation*}

The graphs are shown below:

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From the graph, we see that x=4 and y=2. So the numbers are 4 and 2.

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