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6.04 Simultaneous equations

Worksheet
Graphical method
1

The following graph displays a system of two equations:

State the solution to the system in the form \left(x, y\right).

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2

The following graph displays a system of two linear equations:

Determine the number solutions for the system.

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3

Consider the following linear equations:

\begin{aligned} y &= \dfrac{x}{3} + \dfrac{1}{3} \\ - 8 y &= 8 x + 8 \end{aligned}

a

Determine the x\text{ and }y-intercepts of the line y = \dfrac{x}{3} + \dfrac{1}{3}.

b

Determine the x\text{ and }y-intercepts of the line - 8 y = 8 x + 8.

c

Hence plot the graphs of the 2 equations on the same coordinate plane.

d

State the values of x and y which satisfy both equations.

4

For each of the following pairs of equations:

i

Sketch the graphs of the two equations on the same coordinate plane.

ii

State the values of x and y which satisfy both equations.

a

\begin{aligned} 4 x - 2 y &= 2 \\ - 2 x + 4 y &= 2 \end{aligned}

b

\begin{aligned} y &= 5 x - 7 \\ y &= - x + 5 \end{aligned}

c

x + y = 7 \\ x - y = 3

d

\begin{aligned} x + y &= 7 \\ 4x + 5y &= 32 \end{aligned}

e

\begin{aligned} y &= \dfrac{x}{2} - 3 \\ y &= 6 - x \end{aligned}

f

\begin{aligned} y &= \dfrac{3}{2}x + 4 \\ y &= \dfrac{-1}{2} x + 8 \end{aligned}

Substitution and elimination methods
5

Solve the following systems of equations using the substitution method:

a

\begin{aligned} y &= 5x + 34 \\ y &= 3x + 18 \end{aligned}

b

\begin{aligned} y &= -4x - 17 \\ 3y &= 21x + 147 \end{aligned}

c

\begin{aligned} x &= 4 y - 27 \\ 2 y + x &= 21 \end{aligned}

d

\begin{aligned} x &= 3 y - 21 \\ 8 y + 5 x &= 79 \end{aligned}

e

\begin{aligned} y &= 3 x - 18 \\ x - y &= 10 \end{aligned}

f

\begin{aligned} y &= -5 x + 22 \\ 6 x + y &= 26\end{aligned}

6

Solve the following pairs of simultaneous equations using the elimination method:

a
\begin{aligned} 2x + 5y &= 44 \\ 6 x - 5 y &= -28 \end{aligned}
b
\begin{aligned} - 5 x + 16 y &= 82\\ 25 x - 4 y &= 122 \end{aligned}
c
\begin{aligned} 2 x - 5 y &= 1 \\ -3 x - 5 y &= -39 \end{aligned}
d
\begin{aligned} - 6 x - 2 y &= 46 \\ - 30 x - 6 y &= 246 \end{aligned}
e
\begin{aligned} - \dfrac {x}{4} + \dfrac {y}{5} &= 8 \\ \dfrac {x}{5} + \dfrac {y}{3} &= 1 \end{aligned}
f
\begin{aligned} y + \dfrac {x}{2} &= 3 \\ \dfrac {x}{5} + 3 y &= - 4 \end{aligned}
7

Solve the following pairs of simultaneous equations using an appropriate algebraic method:

a

\begin{aligned} x + y &= 8 \\ 2x - 3y &= 26 \end{aligned}

b

\begin{aligned} y &= 2x + 16 \\ y &= 3x + 21 \end{aligned}

c

\begin{aligned} y &= 4x - 17 \\ x + y &= 38 \end{aligned}

d

\begin{aligned} 3x + y &= 22 \\ y &= -2x + 10 \end{aligned}

e

\begin{aligned} 2x + y &= 5 \\ 5x + 3y &= 9 \end{aligned}

f

\begin{aligned} 11x + 6y &= 27 \\ 7x + 6y &= -9 \end{aligned}

g

\begin{aligned} 6x + 10y &= 59 \\ -4x + 5y &= 19 \end{aligned}

h

\begin{aligned} 18x - 7y &= -64 \\ 3x + 5y &= 14 \end{aligned}

i

\begin{aligned} 5x - 3y &= 46 \\ 3x + 10y &= 4 \end{aligned}

j

\begin{aligned} \dfrac{x}{2} + 3y &= -6 \\ - \dfrac{x}{4} + y &= -7 \end{aligned}

8

Consider the following linear equations:

\begin{aligned} 0.2 x + 0.3 y &= 0.5 \\ 0.5 x + 0.4 y &= 0.2 \end{aligned}

a

Rewrite the system of equations as an equivalent system with the smallest possible integer coefficients.

b

Solve the new system for x and y.

9

Consider the straight line y = m x + c that passes through the two points \left(2, 0\right) and \left(6, - 5 \right).

a

Write a pair of simultaneous equations using the points given.

b

Solve the equations for m and c.

c

Hence state the equation of the straight line that passes through the two points.

Simultaneous equations involving quadratic functions
10

Consider the following system of equations:

\begin{aligned} x^{2} &= y +14 \\ y &= 3x - 16 \end{aligned}

The graph suggests that the two points of intersection are \left(1, - 13 \right) and \left(2, - 10 \right).

Verify that these are the points of intersection by substituting into the equation.

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11

The following functions are displayed on the graph below:

\begin{aligned} y &= x^{2} \\ y &= - 3 x \end{aligned}
a

State the coordinates of the points of intersection.

b

Hence state the solutions to the equation x^{2} = - 3 x.

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Consider the quadratic function y = x^{2} + 4 and the linear function x + y = 4.

a

Sketch the graphs of y = x^{2} + 4 and x + y = 4 on the same coordinate plane.

b

Hence determine the number of solutions to the equation x^{2} + 4 = -x + 4.

c

Find the points of intersection of the two functions.

13

For each of the following pairs of equations:

i

Sketch the graphs of the two equations on the same coordinate plane.

ii

State the coordinates of the points of intersection.

a

\begin{aligned} y &= x^{2} \\ y &= 2 x^{2} - 1 \end{aligned}

b

\begin{aligned} y &= x^{2} + 1 \\ y &= 3 - x^{2} \end{aligned}

14

The function y = x^{2} + 1 is shown on the graph:

a

Determine the number of points of intersection the line y = -2 has with the parabola.

b

State the minimum value of k such that the line y = k has at least one point of intersection with the parabola.

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Vanessa is using algebra to find the points of intersection of a parabola and a straight line. She finds a quadratic equation in the form a x^{2} + b x + c = 0, but when she tries to solve it, she finds that the equation has no solutions.

State whether following graphs could represent the parabola and straight line in this situation.

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16

Find the coordinates of the points where the vertical line x = - 5 intersects the curve \\ y = - 2 x^{2} + x - 12.

17

Find the coordinates of the points of intersection for the following systems of equations:

a

\begin{aligned} y&= x^{2} \\ y&=9 \end{aligned}

b

\begin{aligned} y&=x^{2}-8x \\ y&=-7 \end{aligned}

c
\begin{aligned} y &= 6x^{2}-18x+51 \\ y&=18x+3 \end{aligned}
d
\begin{aligned}x^{2} + y &= 5 \\ 2y &= 3x^{2} - 10 \end{aligned}
e
\begin{aligned} y&=\left( x - 3 \right) \left( x - 2 \right) \\ y &= -13x - 10 \end{aligned}
f
\begin{aligned} y &= - x^{2} - 3 x + 1 \\ y &= - 3 x - 3 \end{aligned}
g
\begin{aligned} y &= x \left(x - 4\right) \\ y &= - 3 x + 12 \end{aligned}
h
\begin{aligned} y &= x^{2} - 7 x + 8 \\ y &= x^{2} + 8 x - 97 \end{aligned}
18

One point of intersection of the curves y = x^{2} - 2 x + 30 and y = k x + 10 occurs at x = 4. Find the value of k.

19

Consider the following pair of parabolas:

\begin{aligned} y &= 2 x^{2} + 11 x + 1 \\ y &= x^{2} + 3 x - 3 \end{aligned}

Find the exact value of the x-coordinates of their points of intersection.

20

The function y = - x^{2} - 2 x + 3 is shown on the graph:

a

State the linear expression that must be added to -x^{2} + x + 2 to form \\ -x^{2} - 2x + 3

b

Hence state the equation of the straight line that would need to be graphed to solve the equation \\ -x^{2} + x + 2 = 0 .

c

Graph the straight line on the same coordinate plane.

d

Hence determine the solutions to \\\ -x^{2} + x + 2 = 0.

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MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

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