Worked examples
The question
Question: Shirts cost $12 and hats cost $8.
Ana spends $96.
Write the equation.
1 Find what we know
What do we know?
Shirts cost $12 each.
Hats cost $8 each.
Ana spends $96 in all.
What do we want?
An equation for this story.
2 Name the unknowns
We don't know how many shirts.
We call that number x.
x = number of shirts
We don't know how many hats either.
It is a different number, so it gets its own letter: y.
y = number of hats
3 Write the equation
One shirt costs $12.
So x shirts cost 12 times x: 12x.
x shirts: 12 × x = 12x
One hat costs $8.
So y hats cost 8 times y: 8y.
y hats: 8 × y = 8y
In words: the shirt money plus the hat money makes all the money.
12x + 8y = 96
So the equation is 12x + 8y = 96.
12x + 8y = 96
Careful!
x + y counts the shirts and hats, not the money.
The money is 12x + 8y.
Careful: not x + y = 96 ✗
4 Check
Let's test the equation with one buy that costs $96.
We try 1 shirt first.
1 shirt costs $12.
$84 is left.
84 divided by 8 is not a whole number.
We can't buy part of a hat.
Next try.
1 shirt: $12, $84 left, 84 ÷ 8 = 10.5 ✗
2 shirts cost $24.
$72 is left.
72 divided by 8 is 9: 9 hats.
2 shirts: $24, $72 left, 72 ÷ 8 = 9 hats ✓
Put x = 2 and y = 9 in: 24 plus 72 is 96.
The equation works!
12 × 2 + 8 × 9 = 24 + 72 = 96
Let's look back.
First: find what we know.
Then: name the unknowns.
Then: write the equation.
Last: check.
Answer Careful: not x + y = 96 ✗
▶ Watch it solved
The question
Question: Roses cost $3 and tulips cost $2.
Eva can spend at most $40.
Write the inequality.
1 Find what we know
What do we know?
Roses cost $3 each.
Tulips cost $2 each.
Eva can spend $40 or less.
What do we want?
An inequality for this story.
2 Name the unknowns
We don't know how many roses.
We call that number x.
x = number of roses
We don't know how many tulips either.
It is a different number, so it gets its own letter: y.
y = number of tulips
3 Write the inequality
One rose costs $3.
So x roses cost 3 times x: 3x.
x roses: 3 × x = 3x
One tulip costs $2.
So y tulips cost 2 times y: 2y.
y tulips: 2 × y = 2y
At most $40 means $40 or less.
That is the sign less than or equal to.
at most $40 = $40 or less: ≤
In words: the rose money plus the tulip money is at most $40.
3x + 2y = 40
So the inequality is 3x + 2y ≤ 40.
3x + 2y ≤ 40
Careful!
At most means 40 or less.
So the sign is less than or equal to, not more than.
Careful: not 3x + 2y ≥ 40 ✗
4 Check
Let's test it with two buys: a small one and a big one.
1 rose and 1 tulip cost $5.
That is at most $40, and the inequality says true.
x = 1, y = 1: 3 + 2 = 5 ≤ 40 true
14 roses cost $42.
That is more than $40, and the inequality says false.
Both match the story.
It works!
x = 14, y = 0: 3 × 14 = 42 not ≤ 40 false
Let's look back.
First: find what we know.
Then: name the unknowns.
Then: write the inequality.
Last: check.
Answer Careful: not 3x + 2y ≥ 40 ✗
▶ Watch it solved
The question
Question: Adult tickets cost $11 and child tickets cost $7.
Ella spends $95.
Write the equation.
1 Find what we know
What do we know?
Adult tickets cost $11 each.
Child tickets cost $7 each.
Ella spends $95 in all.
What do we want?
An equation for this story.
2 Name the unknowns
We don't know how many adult tickets.
We call that number x.
x = number of adult tickets
We don't know how many child tickets either.
It is a different number, so it gets its own letter: y.
y = number of child tickets
3 Write the equation
One adult ticket costs $11.
So x adult tickets cost 11 times x: 11x.
x adult tickets: 11 × x = 11x
One child ticket costs $7.
So y child tickets cost 7 times y: 7y.
y child tickets: 7 × y = 7y
In words: the adult ticket money plus the child ticket money makes all the money.
11x + 7y = 95
So the equation is 11x + 7y = 95.
11x + 7y = 95
Careful!
x + y counts the adult tickets and child tickets, not the money.
The money is 11x + 7y.
Careful: not x + y = 95 ✗
4 Check
Let's test the equation with one buy that costs $95.
We try 1 adult ticket first.
1 adult ticket costs $11.
$84 is left.
84 divided by 7 is 12: 12 child tickets.
1 adult ticket: $11, $84 left, 84 ÷ 7 = 12 child tickets ✓
Put x = 1 and y = 12 in: 11 plus 84 is 95.
The equation works!
11 × 1 + 7 × 12 = 11 + 84 = 95
Let's look back.
First: find what we know.
Then: name the unknowns.
Then: write the equation.
Last: check.
Answer Careful: not x + y = 95 ✗
▶ Watch it solved
