This video will show
how to use the Desmos matrix calculator
to solve a system of two linear equations
with two unknowns, using a matrix equation.
Looking at the system,
notice how both equations are in standard form,
meaning both the X and Y terms are on the left
and the constants are on the right,
and therefore the equations are in the form
where we can write the matrix equation.
To form the matrix equation,
we first form a two by two matrix
using the coefficients of X and Y.
Looking at the first equation,
because the coefficient of X is three,
and the coefficient of Y is four,
the first row is three, four.
Looking at the second equation,
because the coefficient of X is four,
the coefficient of Y is negative two,
the second row is four, negative two.
And then we have times the variable matrix
because the variables are X and Y,
the two by one matrix has elements X, and then Y.
And then this is equal to the constant matrix,
which is another two by one matrix,
looking at the right side of the equations,
notice how the valleys are seven and five,
which give us the elements for the constant matrix.
So now we have our matrix equation, looking at our notes,
the matrix equation is in the form of matrix A,
the coefficient matrix times matrix X, the variable matrix,
which equals matrix B, the constant matrix.
Our goal here is to solve for the matrix X.
To do this, we find the inverse
of matrix A, and then perform the matrix
multiplication A inverse times A,
times X on the left must equal
A inverse times B on the right.
A inverse times A is equal to the identity matrix,
which we identify as I, this gives us
I times X equals A inverse times B,
and I times X is equal to X,
X is equal to A inverse times B.
So now we'll go to the Desmos matrix calculator,
enter matrix A and matrix B,
and then determine A inverse times B.
So going to the Desmos matrix calculator,
we begin by pressing clear all,
and let's enter matrix A, click new matrix.
By default, it's labeled matrix A.
Enter the elements three, right arrow, four,
right arrow, four, right arrow, negative two, enter.
And now let's enter matrix B,
click new matrix. By default it's labeled matrix B.
Matrix B is a two by one matrix.
We need to change the number of columns to one,
enter the elements seven, right arrow, five, enter.
And now let's do two things.
Let's first verify that A inverse times A
does give us the identity matrix.
We press A inverse, A enter,
and notice how we do get the two by two identity matrix.
And now let's find A inverse times B, A inverse B enter,
notice how we're getting decimal
approximations for the elements.
We want the exact values in fraction form.
To do this, move the cursor to the circle
on the far right, this is the convert to fraction button.
If we click this,
it will give us the elements in exact form.
We have 17 elevenths and 13, 20 seconds.
The matrix X is the verbal matrix.
What you get is equal to A inverse times B,
which we now know is a two by one matrix where the elements
are 17 elevenths, and 13 20 seconds,
which indicates the solution is where
X is equal to 17 elevenths,
and Y is equal to 13 20 seconds,
or as an ordered pair, in parentheses,
we have 17 elevenths, comma, 13, 20 seconds.
Because we have one solution,
we know the system is consistent and independent.
Before we go, though,
let's verify this order pair does satisfy
both linear equations. To do this,
we substitute 17 elevenths for X,
and 13 20 seconds for Y in both equations.
So going back to the calculator,
for the first equation, we have three
open parenthesis, 17 divided by 11, right arrow,
close parenthesis, plus four, open parenthesis,
13 divided by 22, right arrow,
close parenthesis, enter.
Notice how this does equal seven,
which verifies the ordered pair
does satisfy the first equation.
And now check the second equation.
You have four, open parenthesis, 17 divided by 11,
right arrow, close parenthesis, minus two,
open parenthesis, 13 divided by 22.
Notice how I made an error in the denominator,
I'm going to go down to the denominator
and delete one of the twos, So now I do have
13 divided by 22 as our fraction.
Right arrow, close parenthesis, enter,
and notice how it does equal five,
which verifies the ordered pair
also satisfies the second equation,
and therefore our solution is correct.
I hope you found this helpful.