When a problem involves two unknown quantities and gives two separate facts about them, let a letter stand for each unknown and turn each fact into an equation. Solving the pair answers the problem. Why two facts are needed: one equation in two unknowns has infinitely many solutions (all the points on a line). A second, independent fact gives a second line, and the two lines meet at one point. For example, let $r be the price of a bag of rice and $v the price of a bottle of cooking oil. **Fact** | **Equation** Five bags of rice and six bottles of oil cost $90. | 5r+6v=90 A bag of rice costs $7 more than a bottle of oil. | r=v+7 Method - Define each unknown in words, with its unit: “Let $r be the price of one bag of rice.” - Write one equation for each fact. Both sides of an equation must be in the same unit (cents with cents, dollars with dollars). - Solve by the most convenient method. - Answer the question in a sentence with units, and check the answer against the original wording, not only against your equations. Reject values that are impossible in the context, such as a negative number of people. Example Substitute r=v+7 into 5r+6v=90: 5(v+7)+6v=90 11v+35=90 v=5 Then r=5+7=12. A bag of rice costs $12 and a bottle of oil costs $5. Check with the words: 5(12)+6(5)=60+30=90 dollars, and 12 is 7 more than 5. Check: “7 more than” means add 7, so r=v+7; a common error is to write r=7v. Another is to mix units, for example writing a price in cents in one equation and in dollars in the other.

Forming equations from a context

Secondary 2

When a problem involves two unknown quantities and gives two separate facts about them, let a letter stand for each unknown and turn each fact into an equation. Solving the pair answers the problem.

Why two facts are needed: one equation in two unknowns has infinitely many solutions (all the points on a line). A second, independent fact gives a second line, and the two lines meet at one point.

For example, let $\$r$ be the price of a bag of rice and $\$v$ the price of a bottle of cooking oil.

Fact	Equation
Five bags of rice and six bottles of oil cost $\$90$.	$5r+6v=90$
A bag of rice costs $\$7$ more than a bottle of oil.	$r=v+7$

Method

Define each unknown in words, with its unit: “Let $\$r$ be the price of one bag of rice.”
Write one equation for each fact. Both sides of an equation must be in the same unit (cents with cents, dollars with dollars).
Solve by the most convenient method.
Answer the question in a sentence with units, and check the answer against the original wording, not only against your equations. Reject values that are impossible in the context, such as a negative number of people.

Example

Substitute $r=v+7$ into $5r+6v=90$:

$\begin{aligned} 5(v+7)+6v&=90 \\ 11v+35&=90 \\ v&=5 \end{aligned}$

Then $r=5+7=12$. A bag of rice costs $\$12$ and a bottle of oil costs $\$5$. Check with the words: $5(12)+6(5)=60+30=90$ dollars, and $12$ is $7$ more than $5$.

Check: “$7$ more than” means add $7$, so $r=v+7$; a common error is to write $r=7v$. Another is to mix units, for example writing a price in cents in one equation and in dollars in the other.

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Fact	Equation
Five bags of rice and six bottles of oil cost \(\$90\).	\(5r+6v=90\)
A bag of rice costs \(\$7\) more than a bottle of oil.	\(r=v+7\)