One-step equations

When you reach for it

An equation such as x + 18 = 52 or 4x = 36 hides a number behind a letter, and exactly one thing has been done to that number: one number added to it, taken from it, multiplied by it or divided into it. Such an equation is a one-step equation, because one move undoes what was done and leaves the letter alone.

Reach for this move whenever you can say in one phrase what happened to the letter: “18 was added”, “it was multiplied by 4”, “three quarters of it was taken”. Many equations that come from a story have this shape. A price times a number of plates is a bill; a summer journey plus some extra minutes is a winter journey; part of a target is the distance swum so far.

You already have both tools. The properties of equality say that doing the same thing to both sides keeps an equation true, as long as you never divide by 0, and inverse operations say which thing to do. Checking a solution, by putting it back in, says whether the answer is right. This lesson puts them to work over three days: adding and subtracting (Part 1), multiplying and dividing, with fractions and decimals (Part 2), and writing the equation from a story (Part 3).

The procedure

Read what was done to the letter. Do the inverse of it to both sides. Check by putting the answer back into the equation you started with. Every case below is that one idea; what changes is which inverse to use and how to do the arithmetic.

Part 1: undoing an addition or a subtraction

A number was added. In x + 18 = 52, 18 was added to x. A bar picture shows what the equation says: a whole of 52 is made of two parts, x and 18.

So x is what is left of 52 once the 18 is taken off. In the language of equations, subtract 18 from both sides, which the subtraction property allows:

On the left, adding 18 and then subtracting 18 leaves x as it was. Check. 34 + 18 = 52.

In general, x + p = q has the solution x = q - p. The number added may come first: p + x is the same as x + p by the commutative property of addition (properties of addition), so the same subtraction works. The letter may also sit on the right: 52 = x + 18 says the same as x + 18 = 52, since an equation read from right to left is still true.

A number was subtracted. In x - 13 = 29, 13 was taken from x and 29 was left. Put the 13 back: add 13 to both sides, by the addition property.

Check. 42 - 13 = 29. In general x - p = q has the solution x = q + p. The answer is larger than q, as it must be: x is what there was before something was taken away.

The letter was subtracted. Sometimes the unknown is the amount taken away, as in p - x = q. Now x is not the number that was changed but the change itself. Adding x to both sides (the addition property again) gives p = q + x, an equation of the first kind, and subtracting q from both sides then gives x = p - q. Example 3 works one through.

Fractions and decimals change nothing about the move, only the arithmetic: line up the decimal points, or rename fractions over a common denominator, exactly as in any sum or difference.

Part 2: undoing a multiplication or a division

The letter was multiplied. 4x means 4 \times x, and 4x = 36 says that four equal parts, each of size x, make 36.

To find one part, share 36 into 4 equal parts: divide both sides by 4, which the division property allows because 4 \neq 0.

Check. 4 \times 9 = 36. The number that multiplies the letter, here 4, is its coefficient. In general px = q has the solution \displaystyle x = \frac{q}{p}, for any coefficient p other than 0. The answer need not be a whole number: 9x = 6 gives \displaystyle x = \frac{6}{9} = \frac{2}{3}, and indeed \displaystyle 9 \times \frac{2}{3} = 6.

The letter was divided. \displaystyle \frac{n}{12} = 2.5 says that n shared into 12 equal parts gives 2.5 in each part. The whole is 12 of those parts, so multiply both sides by 12, by the multiplication property: n = 2.5 \times 12 = 30. Check. 30 \div 12 = 2.5.

The coefficient is a fraction. Take \displaystyle \frac{3}{4}x = 12: three quarters of x is 12.

The bar shows the reasoning. Three equal quarters make 12, so one quarter is 12 \div 3 = 4, and the whole, four quarters, is 4 \times 4 = 16. So x = 16.

The equation does the same in one move. Dividing both sides by \displaystyle \frac{3}{4} is allowed, and dividing by a fraction is the same as multiplying by its reciprocal, the fraction turned upside down, here \displaystyle \frac{4}{3}. Two numbers are reciprocals when their product is 1, and \displaystyle \frac{4}{3} \times \frac{3}{4} = \frac{12}{12} = 1. So multiply both sides by \displaystyle \frac{4}{3}:

On the left the two fractions are multiplied first (regrouping a product is the associative property), giving 1 \times x, which is x. On the right, \displaystyle \frac{4}{3} \times 12 = \frac{48}{3} = 16. Check. \displaystyle \frac{3}{4} \times 16 = \frac{48}{4} = 12.

This covers division too. \displaystyle \frac{n}{12} is \displaystyle \frac{1}{12}n, and the reciprocal of \displaystyle \frac{1}{12} is 12, which is why multiplying by 12 undid it. A whole-number coefficient such as 4 has the reciprocal \displaystyle \frac{1}{4}, and multiplying by \displaystyle \frac{1}{4} is the same as dividing by 4. So one sentence covers all of Part 2: multiply both sides by the reciprocal of the coefficient, which is the same as dividing both sides by the coefficient.

The coefficient is a decimal. In 0.4x = 2.6, divide both sides by 0.4. Dividing by a decimal is easiest after multiplying both numbers by 10, which does not change the quotient: 2.6 \div 0.4 = 26 \div 4 = 6.5. Check. 0.4 \times 6.5 = 2.6. A decimal can also be written as a fraction: \displaystyle 0.4 = \frac{2}{5}, and multiplying both sides by its reciprocal \displaystyle \frac{5}{2} gives the same 6.5.

Part 3: writing the equation from words

A story does not hand you an equation; it hands you a situation. Three decisions turn it into one.

Name the unknown, with its unit. “Let t be the number of lengths Tamar swam” says what the letter counts. “Let t be Tamar” does not, because Tamar is not a number.

Find the sentence that says two amounts are equal. At the swim club, Sima swam 14 more lengths than Tamar, and Sima swam 40. Two expressions now name Sima’s lengths: t + 14, from the comparison, and 40, given outright. Writing each side with words turned into expressions and setting them equal gives

The word “more” is in the story, yet the move that solves the equation is a subtraction: t = 40 - 14 = 26. The word describes how the two amounts compare, not what to do to find the unknown.

Check against the story, then ask whether the answer fits. Putting 26 into t + 14 = 40 checks the arithmetic, but not the equation: if the equation had been written wrongly, its solution would still pass that check. So check in the words. Tamar swam 26 lengths, and 14 more than that is 40, which is what Sima swam. Then ask whether the number can be what it claims to be. A number of lengths is a whole number, and 26 is. Had the equation given 26.5 lengths, the fault would lie in the equation or in the story’s numbers, and the honest answer would say so rather than round.

Worked examples

Examples 1 to 3 go with Part 1, Examples 4 to 6 with Part 2, and Examples 7 and 8 with Part 3.

Boundaries

One operation only. The move undoes exactly one thing. In 3x + 5 = 20 two things were done to x, so two moves are needed, made in reverse order: two-step equations. When the letter appears on both sides it must first be gathered onto one side: equations with variables on both sides.

Never divide by 0. The coefficient must not be 0, and 0 has no reciprocal, since no number times 0 makes 1. The equation 0x = 5 has no solution, because 0 \times x is 0 whatever x is, while 0x = 0 is true for every number.

Numbers below zero. In this lesson every number, every step and every solution is zero or more. That limits the subtraction case: x + p = q has a solution here only when q is at least p, since a number of zero or more, plus p, is never less than p. In the same way p - x = q needs q to be at most p. Equations such as x + 20 = 7 do have a solution among the numbers below zero, and two-step equations solves equations with those numbers.

The solution of the equation is not always an answer to the story. Counts of plates, crossings or lengths must be whole numbers, and a time or a weight cannot be less than zero. When the solution does not fit the story, as in Example 8, say so rather than round it.

Check in the original. Putting the answer into the last line, such as x = 34, proves nothing. It goes into the equation you started with, and for a story, back into the story.

Where this goes next

Every longer equation is solved by one-step moves made in turn. Two-step equations undo two operations in reverse order and bring in numbers below zero, and equations with a bracket come from stories in which a sum is multiplied. The same moves solve inequalities with one twist: one-step inequalities add and subtract exactly as here, but take more care when multiplying or dividing.

Many familiar questions are one-step equations in disguise. Finding the whole when 30\% of it is 18 is solving 0.3w = 18, the move behind percent problems. A proportion such as \displaystyle \frac{x}{6} = \frac{5}{3} is one step: multiply both sides by 6. And a mean with a known total and count is a total shared equally, which is px = q again.

Exercises

Part 1: adding and subtracting

Solve each equation and check your answer.

1. a + 36 = 81
2. r - 28 = 59
3. 104 = h + 67
4. g + 4.6 = 9.25
5. \displaystyle d - 2\frac{1}{3} = 5\frac{1}{2}
6. \displaystyle \frac{7}{10} + v = 1\frac{3}{5}
7. 90 - u = 42.5

Part 2: multiplying and dividing

Solve each equation and check your answer.

8. 9b = 117
9. \displaystyle \frac{j}{7} = 15
10. 12f = 30
11. 6 = 8k
12. \displaystyle \frac{4}{7}w = 20
13. \displaystyle \frac{2}{3}q = \frac{5}{6}
14. 0.8c = 5.6
15. 2.4y = 9
16. \displaystyle \frac{n}{4} = 0.35

Part 3: equations from words

For each, write an equation, solve it, and check the answer in the story.

17. At the swim club, Sima has swum \displaystyle \frac{5}{8} of her target number of lengths for the week. She has swum 30 lengths. What is her target?
18. At the recycling center the glass bin holds 3.5 times as much as the can bin. The glass bin holds 1{,}260 kg. How much does the can bin hold?
19. A ferry left the harbor with some cars on board. At the island 23 cars drove off, and 38 cars stayed on for the next stop. How many cars were on board at the harbor?
20. At a sushi bar every plate costs the same, and 9 plates cost 31.50 dollars. (a) What does one plate cost? (b) Roald says his plates there came to exactly 24 dollars. Can that be right?
21. Quim sorted 15 fewer bins than Roald this week. Quim sorted 34 bins. How many did Roald sort?

Reasoning

22. Explain why dividing both sides of \displaystyle \frac{2}{7}x = 6 by \displaystyle \frac{2}{7} and multiplying both sides by \displaystyle \frac{7}{2} give the same answer, and find it.
23. A ferry ticket costs 6 dollars, and a group’s tickets came to 15 dollars. Write an equation for the number of tickets, solve it, and explain what the solution says about the story.
24. Explain why x - 15 = 9 has a solution among the numbers zero or more, but x + 15 = 9 does not.