Topics
2. Equations
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Middle Years

2.07 Non-linear Simultaneous Equations

Lesson
Worksheet
Practice
Worksheet
Nonlinear simultaneous equations
1

Solve the following systems of equations:

a
\begin{aligned} y &= x^{2} \\ y &= 4 x^{2} -12 \end{aligned}
b
\begin{aligned} y &= x^{2} \\ y &= x^{2} +6x \end{aligned}
c
\begin{aligned} x^{2} + y &= 2 \\ 2 y &= 5 x^{2} - 24 \end{aligned}
d
\begin{aligned} y &= x^{2} - 16 \\ y &= 16 - x^{2} \end{aligned}
e
\begin{aligned} x^2 + y^2 &= 10 \\ x-y &= 4 \end{aligned}
f
\begin{aligned} x^{2} + y^{2} &= 5 \\ - 2 x + 3 y &= 7 \end{aligned}
g
\begin{aligned} y &= x^{2} + 2 \\ y &= 6- x^{2} \end{aligned}
h
\begin{aligned} y &= \left(x - 2\right)^{2} \\ y &= 52 - x^{2} \end{aligned}
i
\begin{aligned} y &= x^{2} - 9 x - 7 \\ y &= x^{2} - x + 25 \end{aligned}
j
\begin{aligned} y &= 2 x^{2} + 10 x - 7 \\ y &= x^{2} + 8 x + 1 \end{aligned}
k
\begin{aligned} y &= - 2 x^{2} - 3 x + 7 \\ y &= 7 x^{2} + 9 x + 11 \end{aligned}
l
\begin{aligned} y &= 2 x^{2} + 4 x + 5 \\ y &= x^{2} - 2 x - 4 \end{aligned}
2

Solve the following systems of equations:

a
\begin{aligned} x^{2} + y^{2} &= 41 \\ x - y^{2} &= - 21 \end{aligned}
b
\begin{aligned} x^{2} + y^{2} &= 4 \\ 8x^2 - 2y^{2} &= 32 \end{aligned}
Applications
3

The sum of the squares of two numbers is 61. The difference of the squares of the two numbers is 11.

a

Set up two equations by letting x and y be the two numbers, where x is the larger of the two numbers.

b

Solve for x.

c

Find all the solutions. Write your answer as ordered pairs.

4

In business, supply and demand functions are ways of expressing the supply or demand of a commodity as a function of its unit price. Market equilibrium is the term that describes when the supply is equal to the demand. Consider the supply and demand functions for a new book bag below, where y is in dollars and x is in thousands of units supplied:

  • Supply function: y = 2 x^{2}

  • Demand function: y = - 4 x + 3 x^{2}

a

Solve the equations for x.

b

Which of these values of x must the equilibrium point be at? Explain your answer.

c

Find the value of y for this value of x.

d

Complete the following statement:

The equilibrium quantity occurs when ⬚ thousand units are being produced, which has a corresponding price of ⬚ dollars per unit.

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