Solve Linear Systems Using Technology

Consider the following $2\times2$2×2 system of linear equations:

$x+y=7$x+y=7
$2x-3y=14$2x−3y=14

Previously we have learned how to solve a simple $2\times2$2×2 system of equations like this, using methods such as the substitution method, elimination method or graphical method. Now we are going to learn how to solve linear equations using technology, such as a CAS calculator.

A CAS calculator can be utilised in two different ways to solve a $2\times2$2×2 linear system of equations. You can use a graphical approach, where you graph both linear lines on a number plane and use the CAS calculator to find the point of intersection, i.e. the $x$x & $y$y values that solve both linear equations. Or you can use the solve function on the CAS calculator to find the point of intersection immediately with no graphing involved.

 

Graphical Approach

First of all, you need to turn on your CAS calculator and go to the graphing section of your CAS calculator. Then you need to enter both linear equations. Depending on what form your linear equation line has, you could enter it straight away as a function of $x$x. So your lines will need to be in the form $y=mx+c$y=mx+c.

If your equations are in standard form $ax+by=c$ax+by=c, you could transpose them so that they can be entered into the calculator as above. However, this isn't compulsory. Your CAS calculator will have the ability to input your equation in standard form.

                              

                            

Once you have entered both of your equations, you may need to adjust the visible domain & range, so that you can see both lines as well as their point of intersection (if it exists). You can do this by zooming in and out of the graph or you can adjust the window settings manually.

    

When you can see the point of intersection, you can then analyse the graph to determine the solutions for $x$x and $y$y. To do this, open the Menu, select Analyse Graph and then select Intersection. Because it is possible for more complicated functions to have several points of intersection, you will need to select $x$x-values to the left and right of your point of intersection. Once this is done, the $x$x and $y$y values for the point of intersection should appear. It may be an exact solution or it could be rounded to a certain number of decimal places, depending on your calculator settings.     

 

As you can see, the point of intersection is $\left(7,0\right)$(7,0). This means that the values $x=7$x=7 and $y=0$y=0 simultaneously solve both equations in the above $2\times2$2×2 system of linear equations.

 

Solving approach

Rather than graph your two linear functions, you can use the CAS calculator to immediately give you the solution to your $2\times2$2×2 linear system of equations. This function can also solve systems of equations that have more variables and more equations, e.g. a $3\times3$3×3 or $4\times4$4×4 system of equations.

   

Start by entering the calculator section of your CAS calculator. You can then use the Menu of your calculator. Look under Algebra or Equations and you should find Solve System of Equations and/or Solve Linear System or Equations. Either one of these options will allow you to solve your system of equations.

You'll need to select how many variables your system has and how many equations you are solving for. For the example above, you only need $2$2 variables and $2$2 equations.

Then all you have to do is enter both of your equations into your CAS calculator. Once you press ENTER, the CAS calculator should immediately give you a solution.

Using this approach, we have found $x=7$x=7 and $y=0$y=0 simultaneously solve both equations in the above $2\times2$2×2 system of linear equations. This matches the answer found above using the graphical approach.

 

How many solutions?

In the example above, we were able to use the CAS calculator to quickly find the solution to our system of linear equations. However, two or more linear equations may not have a point of intersection. In other words, a system of equations doesn't always have the one, unique solution.

Consider the following $2\times2$2×2 system of linear equations.

$x+y=7$x+y=7
$2x+2y=21$2x+2y=21

We can write both equations in gradient-intercept form:

$y=7-x$y=7−x
$y=\frac{21}{2}-x$y=212​−x

Thus, both graphs will have gradient of $-1$−1 but their $y$y-intercepts are $7$7 and $\frac{21}{2}$212​ respectively. Because both lines have the same gradient, these two lines will never intersect one another: they are parallel lines. Therefore, this system of equations has no solution. There are no $x$x and $y$y values that can satisfy both equations.

If you attempt to solve a system of linear equations using technology and you are given no answer or a calculator error, then you know that your system has no solutions. 

Now consider this $2\times2$2×2 linear system of equations.

$x+y=7$x+y=7
$2x+2y=14$2x+2y=14

We can write both equations in gradient-intercept form:

$y=7-x$y=7−x
$y=7-x$y=7−x

We can now see that both of these equations are exactly the same. Both lines have the same gradient and the same $y$y-intercepts. If we were to graph these two lines on a number plane, then we would be graphing the same line twice, one on top of the other. So not only are these also parallel lines, but if a point of intersection is considered to be a point where one lines touches the other, then both of these lines are touching everywhere. That means this system of equations has an infinite number of solutions. There are an infinite number of $x$x & $y$y-values that satisfy both equations.

If you attempt to solve a system of linear equations using technology and you are given an answer that has other parameters such as $n$n or $t$t, this means your system has an infinite amount of solutions and not the one, unique solution.

3 x 3 Systems of linear equations

We have seen so far that pairs of equations in two variables can have $0,1$0,1 or infinitely many solutions. The same can be said for systems of equations in three variables.

These represent planes. Planes can be parallel so that they do not intersect and their equations have no simultaneous solution. Otherwise, a pair of planes intersects along a line and there are infinitely many solutions to their equations. 

We need three planes in order to have a single intersection point. Therefore, we need three equations in the variables $x,$x,$y$y and $z$z in order to have a unique solution.

However, three intersecting planes can be oriented in ways that produce a line of solutions or no solutions at all.  If equations of this kind are entered into your device you are likely to receive an error message or a message that the equations are inconsistent or that they are not independent. (We leave the idea of linear independence unexplained for the moment.)

When working with three or more independent variables, this becomes difficult to solve by graphing, as you are required to work in $3$3-dimensions, or higher! But with the power of technology you can solve systems like these, as above. Before we can obtain a solution to a system of linear equations using a computer algebra system or another device, it may be necessary to set up the equations from information given in ordinary language.

Below is an example of a $3\times3$3×3 system of linear equations, that can be solved using technology.

Worded Example

I have a collection of $124$124 coins. Their denominations are $5$5c, $10$10c and $20$20c. The value of the $5$5c and $10$10c coins together is $\$5.75$$5.75 and the value of the $10$10c and $20$20c coins together is $\$6.90$$6.90. How many of each kind of coin do I have?

The solution involves numbers of coins. So, we choose variables to represent the numbers of the three types of coin. Let $x$x be the number of $5$5c coins, let $y$y be the number of $10$10c coins and let $z$z be the number of $20$20c coins.

From the given information we can write the following equations

$x+y+z=124$x+y+z=124
$0.05x+0.10y+0z=5.75$0.05x+0.10y+0z=5.75
$0x+0.10y+0.20z=6.9$0x+0.10y+0.20z=6.9

This is a system of three equations in the three unknown quantities. Each equation is represented by a plane and, in this case, the three planes intersect at a single point. You should confirm, using your device, that the intersection point has coordinates $(x,y,z)=(81,17,26)$(x,y,z)=(81,17,26). 

It is possible to confirm by hand that these numbers satisfy all three of the equations.

 

Worked Examples

Question 1

Question 2

Question 3