14.05 Matrices and simultaneous equations
Given the following system of linear equations:
4 x + 6 y = 23
8 x + 12 y = 61
Express the system of equations in matrix form.
Calculate the determinant of the coefficient matrix.
Explain the meaning of the determinant.
The following matrix equations represent systems of linear equations:
Write the system of two linear equations represented by the matrix.
Given the following systems of linear equations:
Express the system of equations in matrix form.
Find the determinant of the coefficient matrix.
Solve the system of equations using matrices.
x + 3 y = 23
4 x + 2 y = 32
4 x + y = 49
4 x + 5 y = 101
12 x + 16 y = 32
3 x + 28 y = 20
3 x + 4 y = 24.9
6 x + 7 y = 45.9
5 x + y = 13.33
2 x + 4 y = 9.4
x + y = 1
x + 3 y = 5
Given the following systems of linear equations:
Express the system of equations in matrix form.
Calculate the determinant of the coefficient matrix.
Sketch the lines described by the two equations on the same axes.
Comment on the number of solutions to the system of equations.
- 2 x + 3 y = 2
- 8 x + 12 y = 8
\dfrac{x}{4} - y = - 2
\dfrac{x}{2} - 2 y = 0
Given the following system of linear equations:
3 y = - 6 x + 42
4 y = - 2 x + 20
Express the system of equations in matrix form.
Calculate the determinant of the coefficient matrix.
Solve the system of equations using matrices.
Consider the lines 4 x + 2 y = 8 and 5 x + 5 y = 25:
Express the system of equations in matrix form.
If A is the coefficient matrix in part (a), find A^{ - 1 }.
Solve the system of equations using matrices.
x and y are two numbers , where x > y. The sum of the numbers is 42, and the difference is 16.
Write two equations that describe the relationship between x and y in the form \\a x + b y = c.
Express the system of equations in matrix form.
Calculate the determinant of the coefficient matrix.
Solve the system of equations using matrices.
Given the following system of linear equations:
Express the system of equations in matrix form.
Solve the system of equations using matrix inverse methods and your graphics calculator.
2 x + 4 y + 5 z = 45
5 x + y + z = 37
x + 3 y + 6 z = 42
A quadratic function in the form y =a x^{2}+ bx +c contains the points A \left( - 1 , 5\right),\\ B \left(0, 10\right) \text{ and C} \left(4, - 10 \right).
Construct a system of linear equations to represent this information.
Represent the equations in matrix form.
Use your graphics calculator and matrix inverse methods to determine the values of the coefficients a, b \text{ and } c.
A farmer has two paddocks, one with area x \text{ m}^2, and another with area y \text{ m}^2. The total area of both paddocks is 2600 \text{ m}^2 . The farmer has planted \dfrac{x}{2} \text{ m}^2 and \dfrac{2 y}{3} \text{ m}^2 in corn, and this adds up to 1600 \text{ m}^2.
Write two equations in terms of x and y in the form a x + b y = c where a and b are positive.
Express the system of equations in matrix form.
Evaluate the determinant of the coefficient matrix.
Solve the system of equations using matrices.
A company determines that they spend a total of 63 hours per week on various forms of advertising media. The amount of time spent on print media is 3 hours more than social and video media combined. 2 hours more is spent on video media than social media.
Let s, v and p represent the hours spent on social, video and print media respectively. Construct a set of linear equations to represent this information.
Represent the equations in matrix form.
Use your graphics calculator and matrix inverse methods to determine the number of hours each week spent on the three forms of advertising.
At a school canteen, students can order the Healthy snack which contains 4 pieces of fruit and 2 savoury snacks, or they can order the Yummy snack which contains 1 piece of fruit and 3 savoury snacks.
On a given day the canteen used 260 pieces of fruit and 140 savoury snacks. Let h represent the number of Healthy snacks and y represent the number of Yummy snacks.
Construct an equation for the total amount of fruit used.
Construct an equation for the total amount of savoury snacks used.
Represent the two equations using matrices.
Use your graphics calculator and matrix inverse methods to determine the number of Healthy snacks and Yummy snacks used that day.
An investment fund offers two types of investments, the Blue Chip and the Strong Growth. The Blue Chip requires the purchase of 250 mining company shares and 80 speculative shares. The Strong Growth requires the purchase of 150 mining company shares and 600 speculative shares. The cost of investment in the Blue Chip is \$13\,170 and the cost of investment in the Strong Growth is \$21\,150.
Let m represent the cost of mining company shares and s represent the cost of speculative shares.
Construct an equation for the total cost of the Blue Chip investment.
Construct an equation for the total cost of the Strong Growth investment.
Represent the two equations using matrices.
Use your graphics calculator and matrix inverse methods to determine the cost of the mining company shares and the cost of the speculative shares.
The angles in a triangle are such that the larger angle, z, is double the smaller angle, x. The third angle, y, is 30 \degree less than the larger angle, z.
Construct a system of three linear equations to represent this information.
Represent the equations using matrices.
Use your graphics calculator and matrix inverse methods to determine the angles of the triangle.
The number of cars booked in for window tinting is dependent on several variables. A study determined that for the city of Fairview there is a relationship between the annual window tinting revenue R (in thousands of dollars), the number of cars registered C (in millions), and the annual amount spent on car insurance I (in thousands of dollars). The results for 2017, 2018 and 2019 are tabulated below:
To describe the relationship among these variables, we can use the equation:
R = a + b C + c I, where a, b and c are constants. Construct a system of three linear equations to represent this information.
Write the equations using matrices.
Use your graphics calculator and matrix inverse methods to determine the values of the coefficients a, b, c. Round each coefficient to two decimal places.
| R | C | I | |
|---|---|---|---|
| 2017 | 10\,150 | 113.5 | 890 |
| 2018 | 15\,432 | 142 | 1210 |
| 2019 | 22\,450 | 162.7 | 1494 |
Three different Galactic smart phone models, the 6S, 6V and 6P, are produced at three different factories. The table given represents the number produced in each factory in November 2019. The total costs are \$136\,400, \$137\,200 and \$186\,600 for factory A, B and C respectively.
Use your graphics calculator and matrix inverse methods to determine the cost of making each model.
| 6S | 6V | 6P | |
|---|---|---|---|
| Factory A | 1000 | 1200 | 800 |
| Factory B | 800 | 900 | 1000 |
| Factory C | 1200 | 1500 | 1200 |