NZQA Questions

Simultaneous Equations

Report

Write equationsSolve equations and report in contextStatementDiagram and Geometrical interpretationGeneralisation

Question 1

The gardener wants to grow three types of plants (A, B, and C) in the new garden.The total cost of purchasing 40 of plant A, 60 of plant B, and 100 of plant C is $3 600.The price of each plant A is three dollars more than the price of each plant C.The price of each plant B is six dollars more than twice the price of each plant C.Set up and solve a system of equations to find the cost of each plant C.

Write Equations

The total cost of purchasing 40 of plant A, 60 of plant B, and 100 of plant C is $3 600.40A + 60B + 100C = 3600The price of each plant A is three dollars more than the price of each plant C.A = C + 3The price of each plant B is six dollars more than twice the price of each plant C.B = 6 + 2CSet up and solve a system of equations to find the cost of each plant C.

Solve equations and report result in context

40A + 60B + 100C = 3600A = C + 3B = 6 + 2CUsing calculator plant A costs $15 each; plant B costs $30 each and plant C costs $12 each.

Statement

This system of equations represents 3 planes. The system is independent, and consistent with a unique solution

Diagram and geometrical interpretation

The three planes intersect at a unique point.

Generalisation

Whenever 3 equations are independent (i.e. none of the equations are formed from either one or both of the other equations) and their coefficients are not a combination of the other equations, the system will give a unique solution.

Question 2

The gardener plans to plant another garden using three other types of plants (D, E, and F).Plant D costs $15 per plant, plant E costs $5 per plant, and plant F costs $20 per plant.Seventy-five plants are to be bought, at a total cost of $500.The number of plant E to be purchased is the same as the combined total of the number of plant D and twice the number of plant F.Set up a system of equations for the above information and solve them, clearly justifying your answer and carefully explaining what your result means about the gardener’s plans.

Write equationsThe gardener plans to plant another garden using three other types of plants (D, E, and F).Plant D costs $15 per plant, plant E costs $5 per plant, and plant F costs $20 per plant.15D + 5E + 20F = 500Seventy-five plants are to be bought, at a total cost of $500.D + E + F = 75The number of plant E to be purchased is the same as the combined total of the number of plant D and twice the number of plant F.E = D + 2FSet up a system of equations for the above information and solve them, clearly justifying your answer and carefully explaining what your result means about the gardener’s plans.

Solve equations and report result in context

15D + 5E + 20F = 500 (3D + E + 4F = 100)D + E + F = 75D – E + 2F = 0The calculator does not give a solution,

Show working either

Solve.

Show working or

Solve equations and report in context

There is no solution. Therefore it is not possible for the gardener topurchase 75 plants under the required conditionsand stay within the $500 budget.

Statement

The system of equations is independent and inconsistent i.e. there is no solution.

Diagram and Geometrical interpretation

The system of equations represents 3 planes that form a triangular prism where the lines of intersections of pairs of planes are parallel to each other.

Generalisation

Whenever combinations of multiples of the coefficients (but not the constant terms) of two of the equations form the third equation, provided the planes are not parallel the system forms a triangular prism where no solution is possible.

You don’t need to do this.

Twice the first equation plus the second equation gives the coefficients of the third equation.

Question 3

Two mathematics students, Kyle and Rebecca, were attempting to solve the following system of equations:

Kyle finds that (4,–5,–7) is a solution. Rebecca finds that (–5,13,11) is a solution. Explain how the equations are related, and give a geometric interpretation of this situation.

Solve equations and report in context

Calculator does not give a solution

Show working

Solve.

Solve equations and report in context

There are an infinite number of solutions, so we write the solutions in general form.

Solve equations and report in context

There are an infinite number of solutions, so we write the solutions in general form.

Solve equations and report in context

Kyle finds that (4,–5,–7) is a solution. Rebecca finds that (–5,13,11) is a solution.

Statement

The system of equations represents 3 planes. The system of equations is dependent and consistent with an infinite number of solutions.

Diagram and Geometrical interpretation

Generalisation

Where one of the equations is formed from a combination of multiples of the other two, there will be an infinite number of solutions because they intersect on a line.

1/5 of the first equation, then subtract 1/5 of the second equation.