The document discusses converting systems of linear equations to matrix equations. It provides examples of setting up matrix equations to represent word problems involving linear systems. Specifically, it shows how to take a system of 3 equations with 3 unknowns about ticket sales at a movie theater and set it up as a matrix equation to solve for the number of each type of ticket sold. The matrix equation is then solved using the inverse of the coefficient matrix to find that 22 child tickets, 37 adult tickets, and 11 senior tickets were sold.

2. Learning Intention and Success
Criteria
 Learning Intention: Students will understand that
we can convert a system of linear equations to a matrix
equation.
 Success Criteria: You can solve a worded problem
representing linear equations using matrix equations.

3. Prior Knowledge
 We can solve 𝐴𝑋 = 𝐵 using 𝐴−1
𝐴𝑋 = 𝐵
then we get can pre-multiply on the front by 𝐴−1
, so
𝑋 = 𝐴−1
× 𝐵

4. Converting Linear Equations to
Matrices
If we have the system of equations
𝑎𝑥 + 𝑏𝑦 = 𝑒
𝑐𝑥 + 𝑑𝑦 = 𝑓
we can write this as the matrix equation
𝑎 𝑏
𝑐 𝑑
𝑥
𝑦 =
𝑒
𝑓
To consider why this works, try multiplying it out
𝑎 𝑏
𝑐 𝑑
𝑥
𝑦 =
𝑒
𝑓
𝑎 × 𝑥 + 𝑏 × 𝑦
𝑐 × 𝑥 + 𝑑 × 𝑦
=
𝑒
𝑓
𝑎𝑥 + 𝑏𝑦 = 𝑒
𝑐𝑥 + 𝑑𝑦 = 𝑓

5. Converting Linear Equations to
Matrices
 In general, we can convert a system of linear equations
using the following steps
1) Re-order all of the equations so that they are all have
the variables in alphabetical order and are equal to a
constant.
2) Fill in any missing variables with zero.
3) Create an (𝑛 × 𝑛) matrix with the
coefficients(numbers in front of the letters), call this
A, an (𝑛 × 1) matrix of the variables, call this X, and
an (𝑛 × 1) matrix of the constants, call this B.

6. Example 1
 Convert the system of equations below to a matrix
equation and solve.
3𝑥 + 2𝑦 + 4𝑧 = 16
7𝑧 + 8𝑦 − 𝑥 = 11
−5𝑦 + 6𝑥 − 17 = 0
 Step 1: Reorder
3𝑥 + 2𝑦 + 4𝑧 = 16
−𝑥 + 8𝑦 + 7𝑧 = 11
6𝑥 − 5𝑦 = 17

7. Example 1
 Step 2) Fill in missing with 0s
3𝑥 + 2𝑦 + 4𝑧 = 16
−𝑥 + 8𝑦 + 7𝑧 = 11
6𝑥 − 5𝑦 + 0𝑧 = 17
 Step 3) Create a matrix equation
3 2 4
−1 8 7
6 −5 0
𝑥
𝑦
𝑧
=
16
11
17

8. Example 1
 Step 4) Solve AX=B, using 𝐴−1
3 2 4
−1 8 7
6 −5 0
−1
=
1
17
35 −20 −18
42 −24 −25
−43 27 26
1
17
35 −20 −18
42 −24 −25
−43 27 26
3 2 4
−1 8 7
6 −5 0
𝑥
𝑦
𝑧
=
1
17
35 −20 −18
42 −24 −25
−43 27 26
16
11
17
𝑥
𝑦
𝑧
=
1
17
35 −20 −18
42 −24 −25
−43 27 26
16
11
17
𝑥
𝑦
𝑧
=
2
−1
3

9. Example 1 on the CAS

10. Example 2
A cinema has a child price of $8, an adult price of $14.50
and a senior price of $11.25. At one movie they made a
profit of $836.25. There was a total of 70 people at the
movie. There was double the number of children as
there was seniors. Set up a system of matrix equations
and use it to solve the problem of how many of each type
of ticket they sold.

11. Example 2 solution
A cinema has a child price of $8, an adult price of $14.50 and a senior price of $11.25. At one
movie they made a profit of $836.25. There was a total of 70 people at the movie. There was
double the number of children as there was seniors. Set up a system of matrix equations and
use it to solve the problem of how many of each type of ticket they sold.
 Step 1) Declare your variables.
 Let 𝑥 represent the number of child tickets sold
 Let y represent the number of adult tickets sold
 Let z represent the number of senior tickets sold
 Step 2) Set up linear equations
 Equation for prices: 8𝑥 + 14.5𝑦 + 11.25𝑧 = 836.25
 Equation for total number: 𝑥 + 𝑦 + 𝑧 = 70
 Equation for double: 𝑥 = 2𝑧

12. Example 2 solution
A cinema has a child price of $8, an adult price of $14.50 and a senior price of $11.25. At one
movie they made a profit of $836.25. There was a total of 70 people at the movie. There was
double the number of children as there was seniors. Set up a system of matrix equations and
use it to solve the problem of how many of each type of ticket they sold.
 Step 3) Convert to a system of equations
8𝑥 + 14.5𝑦 + 11.25𝑧 = 836.25
1𝑥 + 1𝑦 + 1𝑧 = 70
1𝑥 + 0𝑦 − 2𝑧 = 0
 Step 4) Convert to a matrix equation
8 14.5 11.25
1 1 1
1 0 −2
𝑥
𝑦
𝑧
=
836.25
70
0
 Step 5) Solve the matrix equation

13. Example 2 solution
A cinema has a child price of $8, an adult price of $14.50 and a senior price of $11.25. At one
movie they made a profit of $836.25. There was a total of 70 people at the movie. There was
double the number of children as there was seniors. Set up a system of matrix equations and
use it to solve the problem of how many of each type of ticket they sold.
 Step 5) Solve the matrix equation
8 14.5 11.25
1 1 1
1 0 −2
𝑥
𝑦
𝑧
=
836.25
70
0
𝑥
𝑦
𝑧
=
8 14.5 11.25
1 1 1
1 0 −2
−1
836.25
70
0
Use CAS at this point
𝑥
𝑦
𝑧
=
22
37
11
 Step 6) State your answer.
Therefore there were 22 child tickets, 37 adult tickets and 11 senior tickets sold.