Solving problems with simultaneous equations

Writing a pair of equations from a worded problem, solving, and answering in context.

Method

Example 1●●

Adult tickets cost $\$15$ and child tickets $\$8$. A group bought $20$ tickets for $\$230$. How many adults?

$a + c = 20$ and $15a + 8c = 230$

$c = 20 - a$: $15a + 160 - 8a = 230$, so $7a = 70$.

There were $10$ adults (and $10$ children).

Example 2●●●

Three pens and two pencils cost $\$8.50$. One pen and two pencils cost $\$4.50$. Find the cost of a pen.

$3p + 2q = 8.5$ and $p + 2q = 4.5$

Subtract: $2p = 4$, so a pen costs $\$2$ (and a pencil $\$1.25$).

Flashcards

Sum and difference

$x + y = S$, $x - y = D$; add to get $2x = S + D$.

Two facts

Two equations

Final step

Answer in words with units.

Practice

$\$15$ adult and $\$8$ child tickets; $20$ tickets cost $\$230$. How many adults?
$3$ pens and $2$ pencils cost $\$8.50$; $1$ pen and $2$ pencils cost $\$4.50$. Cost of a pen?
Two numbers add to $87$ and differ by $29$. Enter the larger, then the smaller.
Using the pens question, what does a pencil cost in dollars?
A farm has chickens and cows: $30$ heads and $84$ legs. How many cows?
Twice a number plus another number is $40$; their difference (first minus second) is $2$. Find the first number.
A rectangle's length is $6$ cm more than its width and its perimeter is $60$ cm. Length?
Plan A costs $\$20 + \$0.50$ per GB, plan B costs $\$40$ flat. At how many GB do they cost the same?