A family owns two businesses that made a combined profit of \$6 million in the previous financial year, with Business B making 2 times as much profit as Business A.
Let x and y be the profits (in millions) of Business A and Business B respectively.
Use the fact that the two businesses made a combined profit of \$6 million to set up an equation involving x and y.
Use the fact that Business B made 2 times as much as Business A to set up another equation relating x and y.
Graph the two equations on the same number plane.
Use the graph to find Business A's profit.
Use the graph to find Business B's profit.
Two twin soccer players are having a contest over who will score the most goals in the season. The twins have scored a combined total of 34 goals so far over the season and Twin A has scored 6 more goals than Twin B.
Let x and y be the number of goals scored by Twin A and Twin B respectively.
Use the fact that the twins have scored a combined total of 34 goals to set up an equation involving x and y.
Use the fact that Twin A has scored 6 more goals than Twin B to set up an equation involving x and y.
Graph the two equations on the same number plane.
Use the graph to find the number of goals Twin A has scored this season.
Use the graph to find the number of goals Twin B has scored this season.
A family returns home from a trip in two separate cars: Car A and Car B. Their distance from home, d, is recorded on the graph below:
Which car got home first?
How long after Car A did Car B get home?
How far from home were they when they started driving?
At what speed was Car A travelling?
After how many hours was the distance between the two cars the greatest?
A band plans to record a demo at a local studio. The cost of renting Studio A is \$250 plus \$50 per hour. The cost of renting Studio B is \$50 plus \$100 per hour.
The cost, y, in dollars of renting the studios for x hours can be modelled by the following linear system:
Studio A: y = 50 x + 250
Studio B: y = 100 x + 50
Graph the lines of both equations on the same graph.
State the values of x and y which satisfy both equations.
Interpret the solution of the equations in context.
Maria and Buzz both walk from their houses to the bus stop every morning. Maria walks 1.5 kilometres further, and together they walk 3.3 kilometres.
Use simultanous equations to find the distance Buzz walks.
When comparing some test results Christa noticed that the sum of her Geography score and Science score was 172, and that their difference was 18. She also noticed that she scored higher for the Geography test.
Let her Geography score be x and her Science score be y.
Write two equations in terms of x and y from the information given.
Use these two equations to find her Geography score.
Find her Science score.
The cost of manufacturing toys, C, is related to the number of toys produced, n, by the formula C = 400 + 2 n. The revenue, R, made from selling n toys is given by R = 4 n.
Sketch the graph of the cost and the revenue on the same number plane.
How many toys need to be produced for the revenue to equal the cost?
State the meaning of the y-coordinate of the point of intersection.
The graph shows the cost C \left(x\right) and the revenue R \left(x\right) from making and selling x units of a certain good:
State which line corresponds to the following functions:
C \left(x\right)
R \left(x\right)
How many units must be sold to reach the break-even point?
For each of the following cost revenue graphs:
State the coordinates of the break-even point.
State the fixed cost.
State which is higher: the cost per unit or the revenue per unit.
The cost for a furniture manufacturer to make an armchair is \$600 per armchair plus a fixed setup cost of \$8500. The armchairs will sell for \$850 each.
Write an expression to represent the cost of manufacturing x armchairs.
Write an expression to represent the revenue generated from the sale of x armchairs.
Find the break-even point.
State the meaning of the break-even point.
The two equations y = 3 x + 35 and y = 4 x represent Laura’s living expenses and income from work respectively.
Find the point of intersection of the two equations.
Sketch both equations on the same number plane.
State the meaning of the point of intersection of the two lines.
The two equations y = 4 x + 400 and y = 6 x represent a company's revenue and expenditure respectively.
Find the point of intersection of the two equations.
Sketch both equations on the same number plane.
State the meaning of the point of intersection of the two lines.
The following graph shows two lines that represent the revenue and cost from selling sandwiches at a local fare:
What is the break even point?
Find the equation of the line representing the revenue, y, from the sale of x sandwiches.
Find the equation of the line representing the cost, y, of making x sandwiches.
Would the stand make a profit or a loss if they sold 70 sandwiches?
For each pair of cost, C \left( x \right), and revenue functions, R \left( x \right), find the number of units x that must be sold to break even.
C \left( x \right) = 20 x + 8100, R \left( x \right) = 32 x
C \left( x \right) = 0.3 x + 1275, R \left( x \right) = 2 x
Given the cost function C \left( x \right) = 0.4 x + 2015 and the revenue function R \left( x \right) = 3 x, find the coordinates of the point of intersection, or the break-even point.
The cost for a furniture manufacturer to make an armchair is \$500 per armchair plus a fixed setup cost of \$3500. The armchairs will sell for \$650 each.
Write an expression to represent the cost of manufacturing x armchairs.
Write an expression to represent the revenue generated from the sale of x armchairs.
Find the x-coordinate of the break-even point.
The sum of two numbers is 56 and their difference is 30.
Set up two equations by letting x and y be the two numbers.
Solve the system of equations.
The length of a rectangle is 12 units more than the width, and the perimeter of the rectangle is 56 units. Let y be the width and x be the length of the rectangle.
Use the fact that the length of the rectangle is 12 units more than the width to set up an equation for x and y.
Use the fact that the perimeter of the rectangle is 56 units to set up an another equation for x and y.
Solve the system of equations.
A mother is currently 10 times older than her son. In 3 years time, she will be 7 times older than her son.
Set up two equations by letting x and y be the present ages of the son and mother respectively.
Solve the system of equations.
Toby's piggy bank contains only 5\text{c} and 10\text{c} coins. It contains 70 coins with a total value of \$3.85.
Set up two equations by letting x and y be the number of 5\text{c} and 10\text{c} coins respectively.
Solve the system of equations.
Oprah invested \$16\,000 in total in two stocks A and B. In one year, the investment in stock A made a 14\% return, while the investment in stock B fell by 6\%. The total annual interest from both stocks was \$700.
Set up two equations be letting x and y be the amounts that she invested in stocks A and B respectively.
Solve the system of equations.
Explain the meaning of the solution of the system of equations.
20 pens and 3 rulers cost \$86 while 4 pens and 15 rulers cost \$46.
Let x and y be the price of the pen and ruler respectively.
Use the fact that 20 pens and 3 rulers cost \$86 to set up an equation.
Use the fact that 4 pens and 15 rulers cost \$46 to set up another equation.
Solve for x to find the price of each pen.
Solve for y to find the price of each ruler.
There are 36 members in a group, and the men outnumber the women by 16. Let x be the number of woman in the group and y be the number of men in the group.
Use the fact that the men outnumber the women by 16 to set up an equation relating x and y.
Use the fact that there are a total of 36 members in the group to form another equation relating x and y.
Solve for x to find the number of women in the group.
Solve for y to find the number of men in the group.
Consider two numbers, represented by x and y, that satisfy the following conditions:
Seven times the first number, x, is added to the second number, y, to get 64.
The difference between three times the first number, x, and the second number, y, is 16.
Write an equation in terms of x and y that satisfies the first condition.
Write an equation in terms of x and y that satisfies the second condition.
Find the value of x and of y.
Christa has \$4000 to invest, and wants to split it up between two accounts:
Account A which earns 7\% annual interest, and
Account B earns 8\% annual interest.
Her target is to earn \$303 total interest from the two accounts in one year.
Write two equations by letting x and y be the amounts, in dollars, that she invests in accounts A and B respectively.
Solve for y, the amount Christa had invested in account B.
Solve for x, the amount Christa had invested in account A.
The number of new jobs created in Wyndburn varies greatly each year. The number of jobs created in 2012 was 260\,000 less than triple the number of jobs created in 2007. The number of jobs created in 2012 was also 480\,000 more than the number of jobs created in 2007.
Let x be the number of jobs created in 2007 and let y be the number of jobs created in 2012.
Set up two equations from the information above.
Solve the system of equations.
How many jobs were created in 2012?
Neil plans to start taking an aerobics class. Non-members pay \$4 per class. Members pay a \$10 fee plus an additional \$2 per class.
The monthly cost, y, of taking x classes can be modelled by the linear system:
Non-members: y = 4x
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Members: y = 2x + 10
Plot the lines of both equations on the same number plane.
State the values of x and y which satisfy both equations.
Interpret these x and y-values for the given context.
There are 28 members in a group, and the men outnumber the women by 14. Let x and y be the number of women and men in the group respectively.
Use the fact that the men outnumber the women by 14 to set up an equation relating x and y.
Use the fact that there are a total of 28 members in the group to form another equation relating x and y.
Hence, solve the system of linear equations using the solving functionality of your CAS calculator, or other tecnology.
For two numbers x and y:
-
Seven times the first number is added to the second number to get 50.
The difference between four times the first number and the second number is 16.
Set up two equations relating x and y. Use x as the first number and y as the second.
Solve the system of linear equations using the solving functionality of your CAS calculator, or other technology.
A mother is currently 7 times older than her son. In 2 years time, she will be 5 times older than her son. Let x and y be the present ages of the son and mother respectively.
Use the fact that the mother is currently 7 times older than her son to set up an equation relating x and y.
Use the fact that the mother will be 5 times older than her son in 2 years time to set up another equation relating x and y.
Hence, solve the system of linear equations using the solving functionality of your CAS calculator, or other technology.
Toby's piggy bank contains only 5\text{c} and 10\text{c} coins. If it contains 48 coins with a total value of \$3.45, find the number of each type of coin.
Let x and y be the number of 5\text{c} and 10\text{c} coins respectively.
Use the fact that the total number of coins is 48 to set up an equation.
Use the fact that the total amount of coins are worth \$3.45 to set up another equation.
Hence, solve the system of linear equations using the solving functionality of your CAS calculator, or other technology.