Word problems solved by a system of equations

When a problem has two unknown quantities and two facts linking them, it is solved by a system of equations:

• Give each unknown a letter, and write what it stands for: “$\displaystyle x$ – the price of …”.
• Write an equation for each fact.
• Solve the system by the method you find easiest.
• Answer the question in words, and check the answer in the story itself.

The Levi family – 2 adults and 3 children – paid 160 ₪ for cinema tickets.
The Cohen family – 1 adult and 2 children – paid 95 ₪.
How much is an adult’s ticket, and how much a child’s?

The solution below calls the price of an adult’s ticket $\displaystyle x$ and the price of a child’s ticket $\displaystyle y$, and writes an equation for each family:

• We isolate $\displaystyle x$ in the second equation: $\displaystyle {x}={95-2y}$.
• We substitute in the first: $\displaystyle {2(95-2y)+3y}={160}$, that is $\displaystyle {190-y}={160}$, so $\displaystyle {y}={30}$.
• $\displaystyle {x}={95-2\cdot30}={35}$.
• Answer: an adult’s ticket costs 35 ₪, a child’s 30 ₪.
• Check in the story: $\displaystyle {2\cdot35+3\cdot30}={160}$ and $\displaystyle {35+2\cdot30}={95}$.

The sum of two numbers is 40.
One number is 8 more than the other.
Find the two numbers.
Write a system of equations and solve it.

Noa’s money box holds 20 coins: 5 ₪ coins and 10 ₪ coins.
Altogether there are 130 ₪ in the box.
How many coins of each kind are in the box?
Write a system of equations and solve it.

At a falafel stand, 3 portions of falafel and 2 bottles of drink cost 69 ₪.
2 portions of falafel and 3 bottles of drink cost 56 ₪.
How much is a portion of falafel, and how much a bottle of drink?
Write a system of equations and solve it.

Two companies rent out bicycles:
At “Galgal” you pay a 30 ₪ sign-up fee plus 5 ₪ an hour.
At “Ofan” you pay 8 ₪ an hour, with no sign-up fee.
Which company is it better to rent from?

We call the number of hours $\displaystyle x$ and the price in shekels $\displaystyle y$. Each company is a line:

By comparison: $\displaystyle {8x}={5x+30}$, $\displaystyle {3x}={30}$, $\displaystyle {x}={10}$, and $\displaystyle {y}={80}$. The lines meet at $\displaystyle (10,80)$ – Ofan’s line is the dashed one:

• Fewer than 10 hours: Ofan’s line is lower – Ofan is cheaper.
• Exactly 10 hours: the same price at both companies, 80 ₪.
• More than 10 hours: Galgal’s line is lower – Galgal is cheaper.

A phone company offers two plans:
Plan A costs 60 ₪ a month, with unlimited calls.
Plan B costs 15 ₪ a month plus 0.5 ₪ for each minute of calls.
On the graph, $\displaystyle x$ is the number of minutes in a month and $\displaystyle y$ the price in shekels; plan B is the dashed line.
Mark the point where the lines meet on the graph.
Write an equation and solve it: how many minutes a month make the two plans cost the same?
Dana talks about two hours a month. Which plan should she choose?

A father is 28 years older than his son.
In 4 years, the father’s age will be 3 times his son’s.
How old are they today?
Write a system of equations and solve it.