Download Lesson 5.4 - james rahn

Solving System of Equations
Using Matrices


In this lesson we will solve systems of
equations using matrices.
This method is similar to the elimination
method, but using matrices may be quicker
because you keep track of equations using a
shorter notation.
5x  3y  1 5 3 1

2 6 50 
2
x

6
y

50



Numerals in the first equation match the numerals in the first row of the matrix.
Numerals in the second equation match the numerals in the second row of the matrix.


The solution to this system of equations is
x = 4 and y = -7.
When we write the solution matrix we want it
to represent the equations, therefore, x = 4
and y = -7 would look like this:
 x  0y  4
1 0 4 

0 1 7 
0
x

y


7



Numerals in the first equation match the numerals in the first row of the matrix.
Numerals in the second equation match the numerals in the second row of the matrix.


In the elimination method you combined
equations and multiplied them by numbers.
In much the same way we can modify the
rows of the matrix by performing row
operations on each number in those rows.
◦ Multiply (or divide) all numbers in a row by a nonzero number
◦ Add all numbers in a row to corresponding
numbers in another row
◦ Add a multiple of the numbers in one row to the
corresponding numbers in another row
◦ Exchange two rows.
Starting matrix
Ending matrix
5 3 1 1 0 4 
2 6 50 0 1 7

 

Solution can be read: x = 4 and y = -7
Solve the system of equations using matrices
 x  2y  3

3x  y  23
Copy the numerals from each equation to form the
matrix
1 2 3 
3 1 23


Add -3 times row 1 to row 2
3 6 9
3
1 2 3 
0 7 14


1 23
0 7 14
1 2 3 
0 7 14


0 1 2
Divide row 2 by 7
1 2 3
0 1 2


Add 2 times row 2 to row 1
0
2
4
1 2 3
1 0 7
1 0 7
0 1 2


Therefore x = 7 and y = 2.


In this investigation you will see how to
combine row operations in your solution
process.
Consider the system of equations:
2 x  y  11

6 x  5y  9


Write the matrix for this system.
Describe how to use row operations to get 0
as the first entry in the second row.
2 1 11 
0 8 24



Use row operations on the matrix from the last step to
get 1 as the second number in row 2
2 1 11
0 1 3 



Use row operations on the matrix from the last step to
get 0 as the second number in row 1
2 0 8
0 1 3


2 0 8
0 1 3



Use row operations on the matrix from the last step to
get 1 as the first number in row 1
1 0 4
0 1 3 



What does the matrix mean? Give the solution to the
system of equations.
x=4 and y = 3.

How do the first three rules for ROW
OPERATIONS IN A MATRIX correspond to
steps we used in the elimination process?
1. Multiply (or divide) all numbers in a row by a
non-zero number
2. Add all numbers in a row to corresponding
numbers in another row
3. Add a multiple of the numbers in one row to the
corresponding numbers in another row
4. Exchange two rows.

On Friday, 3247 people attended the county
fair. The entrance fee for an adult was $5
and for a child 12 and under the fee was $3.
The fair collected $14,273. How many of the
total attendees were adults and how many
were children.

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
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A= number of adult attending
C= number of children attending
Using total attendance: A+C=3247
Using money collected: 5A+3C=14273
1 1 3247 
Set up a matrix
5 3 14,373



Use row operations to simplify the matrix to the
1 1 3247 
read the solution.
5 3 14,373


Add -5 times row 1 to row 2 to get a
new row 2 (-5R1+R2)
Divide row 2 by -1 (R2/-2)
Add -1 times row 2 to row 1 to get a new
row 1 (-1R1+R1)
1 1 3247 
0 2 1962


1 1 3247 
0 1 981 


1 0 2266 
0 1 981 


There were 2266 adults and 981 children attending the fair on
Friday.
Check that 2266 adults and 981 is a solution to both equations:
A+C=3247
5A+3C=14273
2266 +981=3247
3247 = 3247
5(2266)+3(981)=14273
11330+2943=14273

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
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Press 2nd Matrix and Edit Matrix
A. First set the size and the
entries.
Enter the matrix 8 7 1
3 1 4


Return to the home screen and
press 2nd Matrix and choose
Math B: rref(.
Complete the line to say rref([A]).
Press Enter and you will see the
solution matrix for this system.



You leaned to represent a linear system with
a matrix.
You learned to use Row Operations to solve a
system of linear equations.
You learned to solve a matrix on the graphing
calculator.