The sum of two numbers is 56 and their difference is 30. Let x be the larger number and y be the smaller number.
Write two equations for the given information.
Find the two numbers x and y.
For two numbers, x and y:
- Seven times the first number is added to the second number to equal 64.
- The difference between three times the first number and the second number is 16.
Write two equations for the given information.
Find the two numbers x and y.
There are 36 members in a group, and the men outnumber the women by 16. Let x \,be the number of women in the group and y be the number of men in the group.
Write two equations for the given information.
Find the number of men and women in the group.
When comparing some test results Christa noticed that the sum of her Geography test score and Science test score was 172, and that their difference was 18. She scored higher in the Geography test. Let her Geography score be x and her Science score be y.
Write two equations for the given information.
Find Christa's Geography and Science scores.
A mother is currently 10 times older than her son. In 3 years time, she will be 7 times older than her son. Let x and y be the present ages of the son and mother respectively.
Write two equations for the given information.
Find the current ages of the mother and son.
A man is 5 times as old as his son. 4 years ago the man was 9 times as old as his son. Let x and y be the ages of the man and his son respectively.
Write two equations for the given information.
Find the current age of the father and son.
20 pens and 3 rulers cost \$86 while 4 pens and 15 rulers cost \$46. Find the price of each pen and ruler.
Toby's piggy bank contains only 5c and 10c coins. If it contains 70 coins with a total value of \$3.85, find the number of each type of coin.
The number of new jobs created in Wyndburn varies greatly each year:
The number of jobs created in 2019 was 260\,000 less than triple the number of jobs created in 2014.
-
There was an increase of 480\,000 jobs created from 2014 to 2019.
Find the number of jobs created in 2014 and 2019.
Patricia has \$18\,000 to invest, and wants to split it up between two accounts:
Account A earns 8\% and Account B earns 7\% annual interest.
-
Her target is to earn \$1353 total interest from the two accounts in one year.
Let x and y be the amounts, in dollars, that she invests in accounts A and B respectively.
Use the fact that she has \$18\,000 to invest between the two accounts, to write an equation in terms of x and y.
Use the fact that she wants to earn total interest of \$1353, to write an equation in terms of x and y.
Find the amount Patricia invests in each account. Round your answers to the nearest dollar.
A bank loaned out \$12\,000, part of it at a rate of 7\% per year and the rest at the rate of 13\% per year. The interest received for the year totalled \$1158.
Let x and y be the amounts, in dollars, that are loaned at the rates of 7\% and 13\% respectively.
Use the fact that the total amount loaned was \$12\,000, to write an equation in terms of x and y.
Use the fact that the total interest earned was \$1158, to write an equation in terms of x and y.
Find the amount the bank loaned at a rate of 13\% and the amount loaned at 7\%. Round your answers to the nearest dollar.
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.
Write two equations for the given information.
Find the length and width of the rectangle.
Find the value of x and y in the following diagram:
Find the value of x and y in the following diagram:
Hence determine the length and width of the rectangle.
A rectangular garden bed has a perimeter of 13.2 meters. The length is 3.4 metres longer than the width. Find the dimensions of the garden bed.
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, C, of manufacturing x armchairs.
Write an expression to represent the revenue, R, generated from the sale of x armchairs.
Find the number of armchairs that should be produced so that cost of manufacturing equals the revenue generated.
State the cost of manufacturing at the break-even point.
Given the cost function C \left( x \right) = 20 x + 8100 and the revenue function R \left( x \right) = 32 x, find the number of units, x, that must be sold to break even.
The following graph shows two lines that represent the revenue generated f(x) and cost incurred from selling sandwiches at a local fair g(x):
State the coordinates of the break-even point.
Explain why f(x) contains the origin \left(0 , 0 \right) and g(x) does not.
State the fixed costs for the sandwich stand.
Would the sandwich stand make a profit or a loss if they sold 100 sandwiches?
Would the sandwich stand make a profit or a loss if they sold 70 sandwiches?
The two equations y = 4 x + 400 and y = 6 x represent the revenue and expenditure of Company A respectively.
Find the point of intersection of the two equations.
On the same coordinate plane, sketch both equations.
Determine whether a profit or loss is made for the following values of x:
The two equations y = 3 x + 35 and y = 4 x represent Laura’s living expenses and income from work respectively, according to the number of hours per week that she works, x.
Find the point of intersection of the two equations.
On the same coordinate plane, sketch both equations.
Describe the effect on Laura's personal budget if she works the following number of hours per week.
If a system of linear equations in two variables has two graphs that coincide, how many solutions are there to the system?
If a system of linear equations in two variables has two graphs that are parallel lines, how many solutions are there to the system?
State the number of solutions that a system of equations has, if the graphs of the equations are:
Lines that intersect at exactly one point.
Parallel lines.
The same line.
Perpendicular lines.
Consider the following system of equations:
32 x + 12 y =20 \text{ and } 3 y - 5 = -8x
Without solving them, state how many solutions this system of equations has.
Describe each of the following systems of equations as an inconsistent system, or a system with infinitely many solutions:
4 x -9y=5 and - 12 x +27y=5
9 x + 2y = 2 and 27 x + 6y=3
3 x - 3y -5= 0 and x -y-15=0
4x + 2y -7= 0 and 6y = 21 - 12x
2 x -6y=0 and - 5 x +15y = 11
5y = \dfrac{5}{2}x+15 and 2y = x + 6
Determine the number of solutions for the following system of equations:
8 x + 9y=5 \text{ and } \dfrac{8 x}{7} + \dfrac{9 y}{7}=3
Consider the following system of equations:
2 x - 3 y = 9 \text{ and }8 x - 12 y = 24
Graph the two equations on the number plane.
State the number of solutions for the system.
Name the type of system this represents.
Graph the following systems of equations using the CAS calculator and hence:
State the number of solutions to the system of linear equations.
Compare the gradients and y-intercepts of the lines in the system.
- 9 x + 8 y = - 5 and 5 x + 7 y = - 9
- 3 x + y = - 5 and 9 x - 3 y = - 4
Graph the following systems of equations below using the CAS calculator and hence state the values of x and y which satisfy the equations:
y = 4 x and y = 27 - 5 x
4 x - 3 y = - 13 and 4 x + 3 y = 5
Graph the following systems of equations below using the CAS calculator and find the values of x and y which satisfy the system of equations. Write your answer as a pair of coordinates.
y = 2.7 x - 17.41 and y = - 9.8 x + 13.84
\dfrac{1}{3} x + \dfrac{2}{3} y = 1 and \dfrac{1}{2} x + \dfrac{1}{3} y = 7
Solve the following systems of three linear equations using the solve facility of the CAS calculator:
5 x -3y-z= \dfrac{3}{4}
x+2y+4z=\dfrac{29}{10}
4 x + 3y + 2z = \dfrac{67}{20}
2x + 3y-z= -5
4x - 5y +2z= -5
2x +8y +3z= 16
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:
y=4x \text{ and }y=2x + 10
Sketch the graphs of the equations on the same number plane.
State the values of x and y which satisfy both equations.
What do the coordinates of the solution mean in this context?
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 linear system:
y=50 x + 250 \text{ and }y = 100 x + 50
Sketch the lines of the equations on the same number plane.
State the values of x and y which satisfy both equations.
What do the coordinates of the solution mean in this context?
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.
Write an equation for the mother's current age.
Write an equation for the mother's age in 2 years time.
Solve the system of linear equations and hence state the present ages of the mother and son.
Toby's piggy bank contains only 5c and 10c coins. It contains 48 coins with a total value of \$3.45. Let x and y be the number of 5c and 10c coins respectively.
Write an equation for the total number of coins.
Write an equation for the total value of the coins.
Solve the system of linear equations and hence state the number of 5c and 10c coins in Toby's piggy bank.
There are 28 members in a group, and the men outnumber the women by 14. Find the number of men and women in the group.
The function f \left( x \right) = 0.47 x + 8.9 represents the annual bottled water consumption (in billions of litres) of an Australian state. The function g \left( x \right) = - 0.17 x + 14.2 represents the annual soft drink consumption (in billions of litres) for the state. For both functions, x is the number of years since 2015, and these functions are good for the years 2015 to 2029.
Using the system formed by these functions, find x rounded to the nearest whole number.
Hence predict the year in which the bottled water and soft drink consumption will be the same.