Linear Equations · Grade 7

How do you turn a word problem into an equation?

Decide what the unknown stands for and write it out in a full sentence, because everything else follows from that choice. Express the other quantities using the same unknown, translate the sentence that creates an equality into an equation, solve, and then return to the question — it often asks for something other than the unknown itself.

Learning objectives

The decision that settles everything

The hard part is not the solving but the choice of what the unknown will be. Once that is made, the rest of the problem nearly writes itself.

Write the choice as a full sentence: x is Danny's age today. Not just x equals Danny — a whole sentence, including the word today if time matters in the problem.

The simple guide is to choose as the unknown the quantity that the others are described relative to. In an age problem where one person is older than another, it is easier to choose the younger.

Expressing the rest

After the choice, translate every other quantity into an expression in the same unknown. If Danny is five years older than Yossi and Yossi is x, then Danny is x plus five.

In age problems, moving through time adds the same number to every age. In three years Yossi will be x plus three and Danny x plus eight — both aged by the same amount.

Answering what was asked

The equation comes from the sentence that creates an equality: altogether, together, will be twice. That is the sentence joining the two expressions.

After solving, return to the question. If it asked for Danny's age and x was Yossi, the answer is x plus five and not x — and that gap is what turns entirely correct working into a wrong answer.

Finally, check the answer makes sense in the story. A negative age, a price above the budget, or a negative distance is a sign the equation described something else.

Worked examples

  1. Danny is 5 years older than Yossi. Together they are 33. How old is each?

    1. Let x be Yossi's age today
    2. Danny is x plus 5
    3. Together: x plus x plus 5 equals 33
    4. Collect: 2x equals 28

    Answer: Yossi is 14, Danny is 19

  2. A notebook costs 3 times as much as a pencil. Together they cost 16. What does the notebook cost?

    1. Let x be the price of the pencil
    2. The notebook is 3x
    3. Together: x plus 3x equals 16, so 4x equals 16
    4. x equals 4 — but the question asked about the notebook

    Answer: The notebook costs 12

  3. A mother is 36 and her daughter is 8. In how many years will the mother be twice her daughter's age?

    1. Let x be the number of years that pass
    2. In x years: the mother is 36 plus x, the daughter 8 plus x
    3. Twice: 36 plus x equals 2(8 + x)
    4. Expand and gather: 36 plus x equals 16 plus 2x

    Answer: In 20 years. Check: 56 is twice 28

Common mistakes

Answering with the unknown instead of what was asked
The unknown is often the convenient quantity rather than the wanted one. After solving, return to the question and work out what it actually asked for.
Adding the passage of time to only one of the ages
In three years everyone is three years older. Adding to one alone produces an equation that does not describe the story.
Not writing down in words what the unknown stands for
Without that sentence it is easy to mix the two quantities up partway through. One sentence at the start prevents most errors in this topic.

What to remember

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