Simultaneous Linear Equations

Algebra, STD2 A4 EQ-Bank 27

SunPower Solutions is a business that installs solar panels. Fixed costs are $1200. Each panel costs $150.00 to install and generates revenue of $350.00.

The spreadsheet below models the business's costs and revenue for different numbers of panels installed.

Calculate the spreadsheet values for the installation of 4 panels (cells B9, C9, D9) and 6 panels (cells B10, C10, D10). (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
Using the spreadsheet, identify the break-even point and explain what it means for the business. (2 marks)
--- 3 WORK AREA LINES (style=lined) ---
SunPower Solutions has received a large order that will see them make a profit of $4200. Calculate the number of solar panels $(x)$ they will be installing. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

a. $\text{4 panels: TC (B9) = \$1800.00}$

$\text{Revenue (C9) = \$1400.00, Profit/Loss (D9) = }-\$400.00$

$\text{6 panels: TC (B10) = \$2100.00}$

$\text{Revenue (C10) = \$2100.00, Profit/Loss (D10) = \$0.00}$

b. $\text{Break-even = 6 panels, See worked solution}$

c. $27 \text{ panels}$

Show Worked Solution

a. $\text{4 panels:}$

$\text{Total Cost (B9)} = \$1200+4\times\$150 = \$1800.00$

$\text{Revenue (C9)} = 4\times\$350 = \$1400.00$

$\text{Profit/Loss (D9)} = \$1400.00-\$1800.00 = -\$400.00$

$\text{6 panels:}$

$\text{Total Cost (B10)} = \$1200+6\times\$150 = \$2100.00$

$\text{Revenue (C10)} = 6\times\$350 = \$2100.00$

$\text{Profit/Loss (D10)} = \$2100.00-\$2100.00 = \$0.00$

b. $\text{When 6 panels are installed:}$

$\text{Revenue = Total costs = \$2100.00 (breakeven)}$

$\text{This is the point at which the business covers all of its costs and}$

$\text{begins to make a profit.}$

$\text{OR}$

$\text{If fewer than 6 panels are installed the business will make a loss.}$

c. $\text{Let } x = \text{the number of solar panels to be installed.}$

$\text{Profit}$	$ = \text{Revenue}-\text{Total costs}$
$4200$	$ = 350x-(1200+150x)$
$4200$	$= 200x-1200$
$5400$	$=200x$
$x$	$=27$

$\text{SunPower Solutions will install 27 solar panels.}$

Algebra, STD2 A4 EQ-Bank 16

Harmony Arts Festival is a cultural event with fixed costs of $510. Each ticket costs $8.00 to provide and sells for $25.00.

The spreadsheet below models the festival's costs and revenue for different numbers of tickets sold.

Calculate the values for cells C12 and D12 in the spreadsheet. (2 marks)
--- 3 WORK AREA LINES (style=lined) ---
Using the spreadsheet, identify the break-even point and explain what it means for the festival. (2 marks)
--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

a. $\text{C12} = \$1250.00 \quad \text{D12} = \$340.00$

b. $\text{See worked solution}$

Show Worked Solution

a. $\text{C12: Revenue} = 50 \times \$25.00 = \$1250.00$

$\text{D12: Profit/Loss} = \$1250.00-\$910.00 = \$340.00$

b. $\text{When 30 tickets sold:}$

$\text{Revenue = Total costs = \$750.00 (Breakeven)}$

$\text{This is the point at which the festival covers all of its costs}$

$\text{and begins to make a profit.}$

$\text{OR}$

$\text{If less than 30 tickets are sold the festival will make a loss.}$

Algebra, STD2 A4 EQ-Bank 3 MC

Beachside Brew is a cafe with fixed weekly costs of $200. Each cup of coffee costs $1.50 to make and sells for $4.00.

The spreadsheet below models the cafe's weekly costs and revenue for different numbers of cups sold.

How many cups of coffee must be sold each week to break even?

Show Answers Only

$B$

Show Worked Solution

$\text{Break-even occurs when Revenue = Total costs}\ \rightarrow\ \text{(i.e. Profit = \$0)}$

$\text{From the spreadsheet, Revenue = Total costs (\$320.00)}$

$\text{when 80 cups are sold.}$

$\Rightarrow B$

Algebra, STD2 EQ-Bank 25

A local bowling club sold memberships for the new season.

Senior memberships $(s)$ cost $70 each and junior memberships $(j)$ cost $45 each.

On a particular day, a total of 19 memberships were sold, with sales totalling $1030.

Write two equations, in terms of $s$ and $j$, to represent this information. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Determine the exact number of senior and junior memberships sold. (2 marks)

--- 12 WORK AREA LINES (style=lined) ---

Show Answers Only

a. $70s+45j=1030\ …\ (1)$

$s+j=19\ …\ (2)$

b. $s=7,\ \ j=12$

Show Worked Solution

a. $70s+45j=1030\ …\ (1)$

$s+j=19\ …\ (2)$

b. $\text{Rearranging (2) above:}$

$s=19-j$

$\text{Substitute}\ \ s=19-j\ \ \text{into (1):}$

$70(19-j)+45j$	$=1030$
$1330-70j+45j$	$=1030$
$25j$	$=1330-1030=300$
$j$	$=12$

$\text{Substitute}\ \ j=12\ \ \text{into (2):}$

$s=19-12=7$

$\therefore\ \text{7 senior and 12 junior memberships were sold.}$

Algebra, STD2 EQ-Bank 24

A bookshop sold a number of biographies at a weekend sale.

Hardcover biographies $(h)$ sold for $25 each and paperback biographies $(p)$ sold for $12 each.

A total of 11 biographies were sold, with total sales revenue of $210.

Write two equations, in terms of $h$ and $p$, to represent this information. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Determine the number of hardcover biographies and the number of paperback biographies sold. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

$h=6, \ p=5$

Show Worked Solution

a. $25h+12p=210\ …\ (1)$

$h+p=11\ …\ (2)$

b. $\text{Rearranging (2) above:}$

$h=11-p$

$\text{Substitute}\ \ h=9-p\ \ \text{into (1):}$

$25(11-p)+12p$	$=210$
$275-25p+12p$	$=210$
$13p$	$=275-210=65$
$p$	$=5$

$\text{Substitute}\ \ p=5\ \ \text{into (2):}$

$h=11-5=6$

$\therefore\ \text{6 hardcover and 5 paperback biographies were sold.}$

Algebra, STD2 EQ-Bank 32

A school sold tickets to its annual play.

Adult tickets were sold for $8 each and student tickets were sold for $5 each.

A total of 142 tickets were sold, raising $896 in ticket sales.

By writing two equations to represent this information, find the number of adult tickets and the number of student tickets sold. (3 marks)

--- 12 WORK AREA LINES (style=lined) ---

Show Answers Only

$\text{Adult tickets}=62, \ \text{Student tickets}=80$

Show Worked Solution

$\text{Let}\ x=\text{adult tickets},\ y=\text{student tickets}$

$8x+5y=896\ …\ (1)$

$x+y=142\ \ \Rightarrow \ y=142-x\ …\ (2)$

$\text{Substitute}\ \ y=142-x\ \ \text{into (1):}$

$8x+5(142-x)$	$=896$
$8x+710-5x$	$=896$
$3x$	$=896-710=186$
$x$	$=62$

$\text{Substitute}\ \ x=62\ \ \text{into (2):}$

$y=142-62=80$

$\therefore\ \text{62 adult tickets and 80 student tickets were sold.}$

Algebra, STD2 EQ-Bank 30

Eleni is a farmer who sold chickens and ducks at the local market.

Each chicken was sold for $15 and each duck was sold for $9.

She sold a total of 78 birds for a total of $930.

By writing two equations to represent this information, find the number of chickens and the number of ducks Eleni sold. (3 marks)

--- 12 WORK AREA LINES (style=lined) ---

Show Answers Only

$\text{Chickens}=38, \ \text{Ducks}=40$

Show Worked Solution

$\text{Let}\ x=\text{chickens},\ y=\text{ducks}$

$15x+9y=930\ …\ (1)$

$x+y=78\ \ \Rightarrow \ y=78-x\ …\ (2)$

$\text{Substitute}\ \ y=78-x\ \ \text{into (1):}$

$15x+9(78-x)$	$=930$
$15x+702-9x$	$=930$
$6x$	$=930-702=228$
$x$	$=38$

$\text{Substitute}\ \ x=38\ \ \text{into (2):}$

$y=78-38=40$

$\therefore\ \text{Eleni sold 38 chickens and 40 ducks.}$

Algebra, STD2 EQ-Bank 29

Two friends, Adam and Bertha, sold cupcakes for a fundraising event.

Adam sold $x$ cupcakes for $3 each. Bertha sold $y$ cupcakes for $2 each.

Together they sold 113 cupcakes with total sales revenue of $275.

By writing two equations to represent this information, find the number of cupcakes sold by each of the two friends. (3 marks)

--- 12 WORK AREA LINES (style=lined) ---

Show Answers Only

$x=49, \ y=64$

Show Worked Solution

$3x+2y=275\ …\ (1)$

$x+y=113\ \ \Rightarrow \ y=113-x\ …\ (2)$

$\text{Substitute}\ \ y=113-x\ \ \text{into (1):}$

$3x+2(113-x)$	$=275$
$3x+226-2x$	$=275$
$x$	$=275-226=49$

$\text{Substitute}\ \ x=49\ \ \text{into (2):}$

$y=113-49=64$

$\therefore\ \text{Adam sold 49 and Bertha sold 64.}$

Algebra, STD2 A4 2025 HSC 30

It costs $465 to register a passenger car and $350 to register a motorcycle.

Let	$P$	$\ =\ \text{the number of passenger cars, and}$
	$B$	$\ =\ \text{the number of motorcycles}$

Write TWO linear equations that represent the relationship below.

There are 11 times as many passenger cars as motorcycles.
The total registration fees for passenger cars and motorcycles is $494 million. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$\text{Equation 1:}\ P=11B$

$\text{Equation 2:}\ 465P+350B=494\ 000\ 000$

Show Worked Solution

$\text{Equation 1:}\ P=11B$

$\text{Equation 2:}\ 465P+350B=494\ 000\ 000$

♦♦ Mean mark 35%.

Algebra, STD2 A4 2024 HSC 14 MC

Year 11 is making pizzas to raise money for a charity.

The cost $(C)$ and revenue $(R)$ in dollars, when $x$ pizzas are sold are represented by the equations

\begin{aligned}
& C=2.5 x+6 \\
& R=8 x
\end{aligned}

Enough pizzas are sold so that a profit is made.

By how much does the profit increase for each additional pizza sold?

$2.50
$5.50
$6.00
$8.00

Show Answers Only

$B$

Show Worked Solution

$\text {Cost increases by } \$ 2.50 \text { for each extra pizza.}$

$\text {Revenue increases by } \$ 8 \text { for each extra pizza.}$

$\text {Profit (additional) }=8-2.5=\$ 5.50 \text { per pizza}$

$\Rightarrow B$

♦ Mean mark 49%.

Algebra, STD2 A4 2023 HSC 21

Electricity provider $A$ charges 25 cents per kilowatt hour (kWh) for electricity, plus a fixed monthly charge of $40.

Complete the table showing Provider $A$'s monthly charges for different levels of electricity usage. (1 mark)

$\textit{Electricity used in a month (kWh)}$	$\ \ 0\ \ $	$400$	$1000$
$\textit{Monthly charge (\$)}$	$40$		$290$

Provider $B$ charges 35 cents per kWh, with no fixed monthly charge. The graph shows how Provider $B$'s charges vary with the amount of electricity used in a month.

On the grid above, graph Provider $A$'s charges from the table in part (a). (1 mark)
Use the two graphs to determine the number of kilowatt hours per month for which Provider $A$ and Provider $B$ charge the same amount. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
A customer uses an average of 800 kWh per month.
Which provider, $A$ or $B$, would be the cheaper option and by how much? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$\textit{Electricity used in a month (kWh)}$	$\ \ 0\ \ $	$400$	$1000$
$\textit{Monthly charge (\$)}$	$40$	$140$	$290$

c. $\text{400 kWh}$

d. $\text{Provider}\ A\ \text{is cheaper by \$40.}$

Show Worked Solution

$\textit{Electricity used in a month (kWh)}$	$\ \ 0\ \ $	$400$	$1000$
$\textit{Monthly charge (\$)}$	$40$	$140$	$290$

c. $A_{\text{charge}} = B_{\text{charge}}\ \text{at intersection.}$

$\therefore\ \text{Same charge at 400 kWh}$

d. $\text{Cost at 800 kWh:}$

$\text{Provider}\ A: \ 40 + 0.25 \times 800 = $240$

$\text{Provider}\ B: \ 0.35 \times 800 = $280$

$\therefore \text{Provider}\ A\ \text{is cheaper by \$40.}$

Algebra, STD2 A4 2021 HSC 34

In a park the only animals are goannas and emus. Let `x` be the number of goannas and let `y` be the number of emus.

The number of goannas plus the number of emus in the park is 31. Hence `x + y = 31`.

Each goanna has four legs and each emu has two legs. In total the emus and goannas have 76 legs.

By writing another relevant equation and graphing both equations on the grid on the following page, find the number of goannas and the number of emus in the park. (4 marks)

--- 1 WORK AREA LINES (style=lined) ---

Number of goannas = _______________

Number of emus = _________________

--- 0 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{7 goannas, 24 emus}`

Show Worked Solution

`text{Total goanna legs} = 4x`

♦♦♦ Mean mark 29%.

`text{Total emu legs} = 2y`

`4x + 2y`	`= 76`
`2x + y`	`= 38 \ …\ (1)`
`x + y`	`= 31 \ …\ (2)`

`text{Number of goannas} = 7`

`text{Number of emus} = 24`

Algebra, STD2 A4 2020 HSC 24

There are two tanks on a property, Tank A and Tank B. Initially, Tank A holds 1000 litres of water and Tank B is empty.

Tank A begins to lose water at a constant rate of 20 litres per minute.

The volume of water in Tank A is modelled by `V = 1000-20t` where `V` is the volume in litres and `t` is the time in minutes from when the tank begins to lose water.

On the grid below, draw the graph of this model and label it as Tank A. (1 mark)
Tank B remains empty until `t=15` when water is added to it at a constant rate of 30 litres per minute.
By drawing a line on the grid (above), or otherwise, find the value of `t` when the two tanks contain the same volume of water. (2 marks)

--- 2 WORK AREA LINES (style=lined) ---
Using the graphs drawn, or otherwise, find the value of `t` (where `t > 0`) when the total volume of water in the two tanks is 1000 litres. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `text{T} text{ank} \ A \ text{will pass trough (0, 1000) and (50, 0)}`

b. `29 \ text{minutes}`

c. `45 \ text{minutes}`

Show Worked Solution

a. `text{T} text{ank} \ A \ text{will pass trough (0, 1000) and (50, 0)}`

b. `text{T} text{ank} \ B \ text{will pass through (15, 0) and (45, 900)}`

`text{By inspection, the two graphs intersect at} \ \ t = 29 \ text{minutes}`

c. `text{Strategy 1}`

♦♦ Mean mark part (c) 22%.

`text{By inspection of the graph, consider} \ \ t = 45`

`text{T} text{ank A} = 100 \ text{L} , \ text{T} text{ank B} =900 \ text{L} `

`:.\ text(Total volume = 1000 L when t = 45)`

`text{Strategy 2}`

`text{Total Volume}`	`=text{T} text{ank A} + text{T} text{ank B}`
`1000`	`= 1000-20t + (t-15) xx 30`
`1000`	`= 1000-20t + 30t-450 `
`10t`	`= 450`
`t`	`= 45 \ text{minutes}`

Algebra, STD2 A4 2019 HSC 36

A small business makes and sells bird houses.

Technology was used to draw straight-line graphs to represent the cost of making the bird houses `(C)` and the revenue from selling bird houses `(R)`. The `x`-axis displays the number of bird houses and the `y`-axis displays the cost/revenue in dollars.

How many bird houses need to sold to break even? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
By first forming equations for cost `(C)` and revenue `(R)`, determine how many bird houses need to be sold to earn a profit of $1900. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`20`
`96`

Show Worked Solution

a. `20\ \ (xtext(-value at intersection))`

b. `text(Find equations of both lines):`

♦♦ Mean mark 28%.

`(0, 500)\ text(and)\ (20, 800)\ text(lie on)\ \ C`

`m_C = (800-500)/(20-0) = 15`

`=> C = 500 + 15x`

`(0,0)\ text(and)\ (20, 800)\ text(lie on)\ \ R`

`m_R = (800-0)/(20-0) = 40`

`=> R = 40x`

`text(Profit) = R-C`

`text(Find)\ \ x\ \ text(when Profit = $1900:)`

`1900`	`= 40x-(500 + 15x)`
`25x`	`= 2400`
`x`	`= 96`

Algebra, STD2 A4 EQ-Bank 5 MC

A computer application was used to draw the graphs of the equations

`x + y = 4` and `x-y = 4`

Part of the screen is shown.

Which row of the table correctly matches the equations with the lines drawn and identifies the solution when the equations are solved simultaneously?

Show Answers Only

`A`

Show Worked Solution

`text(Line 1:) \ \ x + y = 4`

`text(Line 2:) \ \ x-y = 4`

`text(Intersection at) \ (4, 0).`

`=> A`

Algebra, STD2 A4 EQ-Bank 2 MC

A computer application was used to draw the graphs of the equations

`x-y = 4` and `x + y = 4`

Part of the screen is shown.

What is the solution when the equations are solved simultaneously?

`x = 4, y = 4`
`x = 4, y = 0`
`x = 0, y = 4`
`x = 0, y =-4`

Show Answers Only

`B`

Show Worked Solution

`text(Solution occurs at the intersection of the two lines.)`

`=> B`

Algebra, STD2 A4 2014 HSC 26d

Draw each graph on the grid below and hence solve the simultaneous equations. (3 marks)

`y = 2x + 1`

`x-2y-4 = 0`

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`x=-2,\ y=-3`

Show Worked Solution

`text(Solution is at the intersection:)\ \ x=-2,\ y=-3`

Algebra, STD2 A4 EQ-Bank 22

Two friends, Sequoia and Raven, sold organic chapsticks at the the local market.

Sequoia sold her chapsticks for $4 and Raven sold hers for $3 each. In the first hour, their total combined sales were $20.

If Sequoia sold `x` chapsticks and Raven sold `y` chapsticks, then the following equation can be formed:

`4x + 3y = 20`

In the first hour, the friends sold a total of 6 chapsticks between them.

Find the number of chapsticks each of the friends sold during this time by forming a second equation and solving the simultaneous equations graphically. (5 marks)

Show Answers Only

`text(Sequoia sold 2 and Raven sold 4.)`

Show Worked Solution

`text(Graphing)\ \ 4x + 3y`	`= 20`
`y`	`=-4/3x + 20/3`

`ytext(-intercept) = (0, 20/3)`

`xtext(-intercept)= (5, 0)`

`text(Gradient)\ = -4/3`

`text(S)text(econd equation:)`

`x + y = 6`

`text(From the graph,)`

`text(Sequoia sold 2 and Raven sold 4.)`

Algebra, STD2 A4 EQ-Bank 19

The graph of the line `y = 4-x` is shown.

By graphing `y = 2x + 1` on the grid provided, find the point of intersection of `y = 4-x` and `y = 2x + 1`. (3 marks)

--- 1 WORK AREA LINES (style=lined) ---

Show Answers Only

`(1, 3)`

Show Worked Solution

`text(Graphing)\ y = 2x + 1 :`

`ytext(-intercept = 1)`

`text(Gradient = 2)`

`:.\ text{Point of intersection is (1, 3).}`

Algebra, STD2 A4 EQ-Bank 18

A student was asked to solve the following simultaneous equations.

`y = 2x-8`

`x-4y + 3 = 0`

After graphing the equations, the student found the point of intersection to be `(5,2)`?

Is the student correct? Support your answer with calculations. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(See Worked Solutions)`

Show Worked Solution

`text{Substitute (5, 2) into}\ \ y = 2x-8`

`text(LHS) = y = 2`

`text(RHS) = 2(5)-8 = 2=\ text{LHS}`

`text{Substitute (5, 2) into}\ \ x-4y + 3 = 0`

`text(LHS)= 5-4(2) + 3= 0=\ text(RHS)`

`=> (5,2)\ text(satisfies both equations.)`

`:.\ text(Student is correct.)`

Algebra, STD2 A4 EQ-Bank 17

`y`	`= x + 5`
`y + 2x`	`= 2`

Draw these two linear graphs on the number plane below and determine their intersection. (3 marks)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`(-1,4)`

Show Worked Solution

`text(Table of coordinates:)\ \ y = x + 5`

\begin{array} {|c|c|c|}
\hline \quad x \quad & \ \ -5 \ \ & \ \ -4 \ \ & \ \ -2 \ \ & \quad 0 \quad\\
\hline y & 0 & 1 & 3 & 5 \\
\hline \end{array}

`text(Table of coordinates:)\ \ y + 2x = 2 \ => \ y =-2x + 2`

\begin{array} {|c|c|c|}
\hline \quad x \quad & \ \ -2 \ \ & \ \ -1 \ \ & \quad 0 \quad & \quad 1 \quad\\
\hline y & 6 & 4 & 2 & 0 \\
\hline \end{array}

`text{From graph (and table), intersection occurs}`

`text(at)\ \ (-1,4).`

Algebra, STD2 A4 2018 HSC 27d

The graph displays the cost (`$c`) charged by two companies for the hire of a minibus for `x` hours.

Both companies charge $360 for the hire of a minibus for 3 hours.

What is the hourly rate charged by Company A? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
Company B charges an initial booking fee of $75.

Write a formula, in the form of `c = mx + b`, for the cost of hiring a minibus from Company B for `x` hours. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
A minibus is hired for 5 hours from Company B.

Calculate how much cheaper this is than hiring from Company A. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `$120`

b. `c = 95x + 75`

c. `$50`

Show Worked Solution

a. `text(Hourly rate)\ (A)= 360 ÷ 3= $120`

b. `m = text(hourly rate)`

`text(Find)\ m,\ text(given)\ c = 360,\ text(when)\ \ x = 3\ \ text(and)\ \ b = 75`

`360`	`= m xx 3 + 75`
`3m`	`= 285`
`m`	`= 95`

`:. c = 95x + 75`

c.	`text(C)text(ost)\ (A)`	`= 120 xx 5 = $600`
	`text(C)text(ost)\ (B)`	`= 95 xx 5 + 75 = $550`

`:.\ text(The hiring cost for Company)\ B\ text(is $50 cheaper.)`

Algebra, STD2 A4 EQ-Bank 28

Penny is a baker and makes meat pies every day.

The cost of making `p` pies, `$C`, can be calculated using the equation

`C = 675 + 3.5 p`

Penny sells the pies for $5.75 each, and her income is calculated using the equation

`I = 5.75 p`

On the graph, draw the graphs of `C` and `I`. (2 marks)

--- 0 WORK AREA LINES (style=lined) ---
On the graph, label the breakeven point and the loss zone. (2 marks)

--- 0 WORK AREA LINES (style=lined) ---

Show Answers Only

a. & b.

Show Worked Solution

b. `text(Loss zone occurs when)\ C > I,\ text(which is shaded in the diagram above.)`

Algebra, STD2 A4 EQ-Bank 26

Temperature can be measured in degrees Celsius (`C`) or degrees Fahrenheit (`F`).

The two temperature scales are related by the equation `F = (9C)/5 + 32`.

Calculate the temperature in degrees Fahrenheit when it is `-20` degrees Celsius. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
The following two graphs are drawn on the axes below:

`F = (9C)/5 + 32` and `F = C`

Explain what happens at the point where the two graphs intersect. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `-4^@F`

b. `text(The two graphs intersect at a temperature where)`

`text(Celsius and Fahrenheit are the same.)`

Show Worked Solution

a. `F= (9xx(-20))/5 + 32=-4^@F`

b. `text(The two graphs intersect at a temperature where)`

`text(Celsius and Fahrenheit are the same.)`

Algebra, STD2 A4 2017 HSC 17 MC

The graph of the line with equation `y = 6-2x` is shown.

When the graph of the line with equation `y = x + 3` is also drawn on this number plane, what will be the point of intersection of the two lines?

`(0, 6)`
`(1, 4)`
`(2, 2)`
`(3, 0)`

Show Answers Only

`B`

Show Worked Solution

`text(Method 1 – Graphically)`

`text(From graph, intersection at)\ (1,4)`

`=>B`

`text(Method 2 – Algebraically)`

`y`	`= 6-2x`	`…\ (1)`
`y`	`= x + 3`	`…\ (2)`

`text{Substitute (2) into (1)}`

`x + 3`	`= 6-2x`
`3x`	`= 3`
`x`	`= 1`

`text(When)\ \ x = 1,\ y = 4`

`=>B`

Algebra, STD2 A4 2015 HSC 28f

A charity seeks to raise money by telephoning people at random from a call centre and asking them to donate.

Over the years, this charity has found that the amount of money raised `($A)` is related to the number of telephone calls made `(n)`. A graph of this relationship is shown.

It costs the charity $2100 per week to run the call centre. It also costs an average of 50 cents per telephone call.

Write an equation to represent the total cost, `C`, of running the call centre for a week in which `n` phone calls are made. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
By graphing this equation on the axes above, determine the number of phone calls the charity needs to make in order to break even. (2 marks)

--- 1 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `C = $2100 + $0.50n`

b. `text(700 calls)`

Show Worked Solution

a. `C = $2100 + $0.50n`

♦ Mean marks of 48% and 32% for parts (a) and (b) respectively.

`text(From the above graph, the charity needs to)`

`text(make 700 calls to break even.)`

Algebra, STD2 A4 2005 HSC 28b

Sue and Mikey are planning a fund-raising dance. They can hire a hall for $400 and a band for $300. Refreshments will cost them $12 per person.

Write a formula for the cost ($C) of running the dance for `x` people. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

The graph shows planned income and costs when the ticket price is $20.

Estimate the minimum number of people needed at the dance to cover the costs. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
How much profit will be made if 150 people attend the dance? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Sue and Mikey plan to sell 200 tickets. They want to make a profit of $1500.

What should be the price of a ticket, assuming all 200 tickets will be sold? (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `700 + 12x`

b. `text(Approximately 90)`

c. `$500`

d. `$23`

Show Worked Solution

a. `$C= 400 + 300 + (12 xx x)= 700 + 12x`

b. `text(Using the graph intersection:)`

`text(Approximately 90 people are needed to cover the costs.)`

c. `text(If 150 people attend)`

`text(Income)= 150 xx $20= $3000`

`text(C)text(osts)= 700 + (12 xx 150)= $2500`

`:.\ text(Profit)= 3000-2500= $500`

d. `text(C)text(osts when)\ x = 200:`

`C=700 + (12 xx 200)= $3100`

`text(Income required to make $1500 profit)`

`= 3100 + 1500= $4600`

`:.\ text(Price per ticket)= 4600/200= $23`

Algebra, STD2 A4 2004 HSC 16 MC

George drew a correct diagram that gave the solution to the simultaneous equations

`y = 2x-5` and `y = x + 6`.

Which diagram did he draw?

Show Answers Only

`D`

Show Worked Solution

`text(By elimination)`

`y = 2x-5\ \ text(cuts the y-axis at)\ -5`

`:.\ text(Cannot be A or B)`

`y = x + 6\ \ text(cuts the y-axis at)\ 6`

`text(AND has a positive gradient)`

`:.\ text(Cannot be C)`

`=> D`

Algebra, STD2 A4 EQ-Bank 20

Fiona and John are planning to hold a fund-raising event for cancer research. They can hire a function room for $650 and a band for $850. Drinks will cost them $25 per person.

Write a formula for the cost ($C) of holding the charity event for `x` people. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
The graph below shows the planned income and costs if they charge $50 per ticket. Estimate the number of guests they need to break even. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
How much profit will Fiona and John make if 80 people attend their event? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `$C = 1500 + 25x`

b. `60`

c. `$500`

Show Worked Solution

a. `text{Fixed Costs} = 650 + 850= $1500`

`text(Variable C) text(osts) = $25x`

`:.\ $C = 1500 + 25x`

b.	`text(From the graph)`
	`text(C) text(osts = Income when)\ x = 60`
	`text{(i.e. where graphs intersect)}`

c. `text(When)\ \ x = 80:`

`text(Income)= 80 xx 50$4000`

`$C= 1500 + 25 xx 80= $3500`

`:.\ text(Profit)= 4000-3500= $500`

Algebra, STD2 A4 2011 HSC 20 MC

A function centre hosts events for up to 500 people. The cost `C`, in dollars, for the centre
to host an event, where `x` people attend, is given by:

`C = 10\ 000 + 50x`

The centre charges $100 per person. Its income `I`, in dollars, is given by:

`I = 100x`

How much greater is the income of the function centre when 500 people attend an event, than its income at the breakeven point?

`$15\ 000`
`$20\ 000`
`$30\ 000`
`$40\ 000`

Show Answers Only

`C`

Show Worked Solution

♦ Mean mark 50%
COMMENT: Students can read the income levels directly off the graph to save time and then check with the equations given.

`text(When)\ x=500,\ I=100xx500=$50\ 000`

`text(Breakeven when)\ \ x=200\ \ \ text{(from graph)}`

`text(When)\ \ x=200,\ I=100xx200=$20\ 000`

`text(Difference)=50\ 000-20\ 000=$30\ 000`

`=> C`

Algebra, STD2 A4 2010 HSC 24b

Ashley makes picture frames as part of her business. To calculate the cost, `C`, in dollars, of making `x` frames, she uses the equation `C=40+10x`.

She sells the frames for $20 each and determines her income, `I`, in dollars, using the equation `I=20x`.

Use the graph to solve the two equations simultaneously for `x` and explain the significance of this solution for Ashley's business. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`x=4`

Show Worked Solution

`text(From the graph, intersection occurs at)\ x=4`

♦ Mean mark 36%.
MARKER’S COMMENT: The intersection on the graph is the same point at which the two simultaneous equations are solved for the given value of `x`.

`=>\ text(Break-even point occurs at)\ x=4`

`text(i.e. when 4 frames sold)`

`text(Income)`	`=20xx4=$80\ \ text(is equal to)`
`text(C)text(osts)`	`=40+(10xx4)=$80`

`text(If)\ <4\ text(frames sold)=>\ text(LOSS for business)`

`text(If)\ >4\ text(frames sold)=>\ text(PROFIT)`

\(\text{Profit}\)	\( = \text{Revenue}-\text{Total costs}\)
\(4200\)	\( = 350x-(1200+150x)\)
\(4200\)	\(= 200x-1200\)
\(5400\)	\(=200x\)
\(x\)	\(=27\)

\(70(19-j)+45j\)	\(=1030\)
\(1330-70j+45j\)	\(=1030\)
\(25j\)	\(=1330-1030=300\)
\(j\)	\(=12\)

\(25(11-p)+12p\)	\(=210\)
\(275-25p+12p\)	\(=210\)
\(13p\)	\(=275-210=65\)
\(p\)	\(=5\)

\(8x+5(142-x)\)	\(=896\)
\(8x+710-5x\)	\(=896\)
\(3x\)	\(=896-710=186\)
\(x\)	\(=62\)

\(15x+9(78-x)\)	\(=930\)
\(15x+702-9x\)	\(=930\)
\(6x\)	\(=930-702=228\)
\(x\)	\(=38\)

\(3x+2(113-x)\)	\(=275\)
\(3x+226-2x\)	\(=275\)
\(x\)	\(=275-226=49\)

\(\textit{Electricity used in a month (kWh)}\)	\(\ \ 0\ \ \)	\(400\)	\(1000\)
\(\textit{Monthly charge (\$)}\)	\(40\)		\(290\)

\(\textit{Electricity used in a month (kWh)}\)	\(\ \ 0\ \ \)	\(400\)	\(1000\)
\(\textit{Monthly charge (\$)}\)	\(40\)	\(140\)	\(290\)

\(\textit{Electricity used in a month (kWh)}\)	\(\ \ 0\ \ \)	\(400\)	\(1000\)
\(\textit{Monthly charge (\$)}\)	\(40\)	\(140\)	\(290\)