Math 6 · 6.EE.B.7 · 7.EE.B.4A

Inverse operations and the properties of equality

An equation says two amounts are equal, but one of them is hidden behind a letter. What can you do to the equation without breaking that equality?

You need

What a solution is, and checking oneMath 6

Unlocks

One-step equationsMath 6 Properties of equality and congruenceGeometry

Where you are 11 in Linear equations

Algebra9

Linear equations11

What a solution is, and checking one

Inverse operations and the properties of equality

One-step equations Two-step equations Equations from proportions and percents ▾ Show all 11 (6 more)

Why this exists

A solution of an equation is a number that makes it true, and any guess can be tested by putting it back in: checking a solution. Testing says whether a guess is right. It does not say what to guess. For x + 6 = 19 a good guess comes quickly; for 27n = 1{,}053 or y + 3.85 = 11.2 it does not.

What is needed is a way to change an equation, one move at a time, into one so plain that it reads off its own answer, such as x = 13, without any move changing the answer. Two ideas do this. Inverse operations say which move frees the letter. The properties of equality say why the move is allowed: whatever is done to one side is done to the other.

Definition

Inverse operations. Two operations are inverse operations when each undoes the other: do one, then the other, and you are back where you started.

Adding a number and subtracting the same number are inverse operations: (n + 6) - 6 = n, and (n - 6) + 6 = n.
Multiplying by a number and dividing by the same number are inverse operations, as long as that number is not 0: (n \times 4) \div 4 = n, and (n \div 4) \times 4 = n. This is how multiplication and division undo each other.

Properties of equality. An equation says that two expressions name the same number. If a = b is true, then each of these is true too, for any number c:

Property	If a = b, then
Addition property	a + c = b + c
Subtraction property	a - c = b - c
Multiplication property	a \times c = b \times c
Division property	a \div c = b \div c, when c \neq 0

In words: add, subtract, multiply or divide both sides by the same number, but never divide by 0.

Two equations are equivalent when they have exactly the same solutions. A property of equality replaces one equation with an equivalent, simpler one.

Intuition

A pan balance is a picture of an equation. When the two pans hold the same weight the beam is level, just as = says the two sides are the same number.

Faye puts her suitcase on the left pan with a 6 kg weight beside it. The right pan holds a 19 kg weight, and the beam is level. Writing x for the suitcase’s weight in kilograms, the scale says x + 6 = 19.

Figure 1Lift 6 kg off each pan and the scale stays level. What is left reads off the answer.

Lift 6 kg off each pan. Both pans lost the same weight, so the beam stays level, and what is left says x = 13. Taking 6 away undid the 6 that was added; taking it from both pans kept the balance. Changing one pan only would not.

Figure 2Add 2 kg to one pan only and the scale tips: the two sides are no longer equal.

Multiplying and dividing work the same way. Three identical crates of apples balance three 15 kg weights, so if each crate weighs c kilograms, 3c = 45. Split each pan into three equal shares. Equal amounts split into the same number of equal shares give equal shares, so one crate balances 15 kg and c = 15.

Figure 3Split each pan into three equal shares. One share balances one share.

Results

Same to both sides. Add, subtract, multiply or divide both sides of a true equation by the same number and the new equation is true too. The one exception: never divide by 0.

Choosing the move. To get the letter alone, do the inverse of whatever was done to it, to both sides. Here ax means a \times x.

The equation	Undo it with	License
x + a = b	subtract a from both sides	subtraction property
x - a = b	add a to both sides	addition property
ax = b	divide both sides by a	division property
\dfrac{x}{a} = b	multiply both sides by a	multiplication property

Equivalent equations. Each of these moves, with a number other than 0 for multiplying or dividing, gives an equivalent equation: no solution is lost and none is gained.

Why it is true

Why the properties hold. If a = b, then a and b are two names for one number. Adding c to that number gives one answer, whichever name was used, so a + c = b + c. Subtracting, multiplying and dividing are the same: one number, one operation, one result.

Why the solutions do not change. Take x + 6 = 19. If some number makes it true, subtracting 6 from both sides keeps it true, so that number makes x = 13 true. Going back, if a number makes x = 13 true, adding 6 to both sides keeps it true, so it makes x + 6 = 19 true. Every move can be undone by its inverse, so the two equations are true for exactly the same numbers. This is why inverse operations and the properties of equality belong together.

Why not divide by 0. 7 \div 0 would be the number that, multiplied by 0, gives 7. Every number times 0 is 0, so no number works. 0 \div 0 would be a number that, times 0, gives 0; every number does that, so there is no single answer. Either way dividing by 0 has no value, so “divide both sides by 0” is not a move.

Worked examples

Example 1

Undoing an addition

On a balance, Halle’s suitcase and a 7 kg weight on the left pan balance 31 kg of weights on the right pan. Write an equation and find the weight of the suitcase.

Solution

Call the suitcase’s weight s kilograms. The left pan holds s + 7, the right pan holds 31, and the beam is level:

s + 7 = 31

Seven was added to s, and subtracting 7 undoes that. The subtraction property allows taking 7 from both sides, which on the scale means lifting 7 kg off each pan:

\begin{aligned} s + 7 - 7 &= 31 - 7 \\ s &= 24 \end{aligned}

The suitcase weighs 24 kg. Check. 24 + 7 = 31.

Example 2

Undoing a multiplication

Imre picks four crates of apples, all the same weight. Together they weigh 74 kg. How much does one crate weigh?

Solution

Call one crate’s weight c kilograms. Four crates weigh 4 \times c, written 4c:

4c = 74

The letter was multiplied by 4, and dividing by 4 undoes that. The division property allows dividing both sides by 4, because 4 \neq 0:

\begin{aligned} \frac{4c}{4} &= \frac{74}{4} \\ c &= 18.5 \end{aligned}

One crate weighs 18.5 kg. Check. 4 \times 18.5 = 74.

Example 3

Undoing a division

Mabel shares the apples from one crate equally into 6 bags, and each bag holds 2.25 kg. How heavy were the apples in the crate?

Solution

Call the weight of the apples w kilograms. Sharing into 6 equal bags divides it by 6:

\frac{w}{6} = 2.25

Multiplying by 6 undoes dividing by 6, and the multiplication property allows multiplying both sides by 6:

\begin{aligned} \frac{w}{6} \times 6 &= 2.25 \times 6 \\ w &= 13.5 \end{aligned}

The crate held 13.5 kg of apples. Check. 13.5 \div 6 = 2.25.

Example 4

Naming the license for each step

Each equation becomes the next by one move. Name the move and the property of equality that allows it. (a) From y - 9 = 14 to y = 23. (b) From \displaystyle 5 = \frac{k}{8} to 40 = k. (c) From \displaystyle \frac{2}{3} + p = 5 to \displaystyle p = 4\frac{1}{3}.

Solution

(a) 9 was subtracted from y, so add 9 to both sides: y - 9 + 9 = 14 + 9, which is y = 23. This is the addition property.

(b) k was divided by 8, so multiply both sides by 8: 5 \times 8 = 40 and \displaystyle \frac{k}{8} \times 8 = k. This is the multiplication property. The letter may end on either side; 40 = k says the same as k = 40.

(c) \displaystyle \frac{2}{3} was added to p, so subtract \displaystyle \frac{2}{3} from both sides. On the left, the order of adding does not matter (the commutative property), so \displaystyle \frac{2}{3} + p - \frac{2}{3} = p. On the right, \displaystyle 5 - \frac{2}{3} = \frac{15}{3} - \frac{2}{3} = \frac{13}{3} = 4\frac{1}{3}. This is the subtraction property.

Example 5

Which moves keep the equation true

The equation m + 12 = 30 has the solution m = 18. For each move, decide whether a property of equality allows it, then test whether 18 still solves the new equation. (a) Add 5 to both sides. (b) Multiply both sides by 2. (c) Subtract 12 from the left side only. (d) Subtract 12 from the left side and 10 from the right side.

Solution

(a) Allowed, by the addition property. The new equation is m + 17 = 35, and 18 + 17 = 35, so 18 still solves it.

(b) Allowed, by the multiplication property. The whole left side is doubled, giving 2(m + 12) = 60, and 2 \times 30 = 60, so 18 still solves it.

(c) Not allowed: one side changed and the other did not. The result is m = 30, and 18 does not solve it. On a balance, the beam tips.

(d) Not allowed: the two sides changed by different amounts. The result is m = 20, and 18 does not solve it.

Every allowed move kept the solution. But (a) and (b), though allowed, bring m no closer to standing alone. The property says what is allowed; the inverse operation says what is useful.

Example 6

Why dividing by zero is ruled out

The equation 3 \times 0 = 5 \times 0 is true. Show what would follow if dividing both sides by 0 were allowed.

Solution

Both sides equal 0, so the equation is true. If dividing by 0 undid multiplying by 0, the way dividing by 4 undoes multiplying by 4, then dividing both sides by 0 would leave 3 = 5, which is false. A move that can turn a true equation into a false one cannot be allowed.

The fault lies in dividing by 0 itself. Every number times 0 is 0, so nothing can tell which number was multiplied by 0. That is why the division property says “not 0”.

Example 7

Undoing two operations

Perla thinks of a number, multiplies it by 3, then adds 8, and gets 59. Find her number.

Solution

Undo the operations in reverse order. The last thing she did was add 8, so undo that first: 59 - 8 = 51. Before that she multiplied by 3, so undo it: 51 \div 3 = 17.

The same work is an equation solved with two properties. Her steps give 3n + 8 = 59. Subtract 8 from both sides (subtraction property) to get 3n = 51, then divide both sides by 3 (division property) to get n = 17.

Check. 3 \times 17 + 8 = 51 + 8 = 59. Her number is 17.

The point here is only that each step is one property of equality. Solving equations like this is its own lesson, two-step equations.

Boundaries

Never divide by 0. Adding, subtracting and multiplying are allowed with any number. Dividing is allowed with any number except 0, because division by 0 has no answer.

Multiplying by 0 keeps the equation true but throws the solution away. It turns every equation into 0 = 0, which is true for every number, and it cannot be undone, since undoing it would mean dividing by 0. Solving only multiplies or divides by a number other than 0.

A move applies to a whole side. A side is the whole expression on one side of the = sign. Doubling the left side of m + 12 = 30 gives 2(m + 12), every part doubled.

These are properties of equations. For inequalities, which use < and >, adding and subtracting work the same way, but multiplying and dividing need more care once numbers below zero are in play: one-step inequalities.

Where this goes next

These properties are the license for every step of solving an equation from here on. One-step equations puts them to work on all four operations, with fractions, decimals and equations written from a story. When two operations have been done to the letter, as in Example 7, the moves come in reverse order: two-step equations. When the letter appears on both sides, the subtraction property takes away a letter term rather than a number: equations with variables on both sides.

In geometry the same properties justify steps in a proof: properties of equality and congruence.

Exercises

Inverse operations

What operation undoes “add 17”?
What operation undoes “divide by 9”?
Start with 26, multiply by 11, then divide by 11. What number do you end with, and why?
Write (n - 35) + 35 more simply.
Which of these pairs are inverse operations? (a) Add 2.5, then subtract 2.5. (b) Multiply by 3, then divide by 6. (c) Multiply by 5, then divide by 5. (d) Add 8, then subtract 4.

Naming the property

Each equation becomes the next by one move. Name the move and the property of equality that allows it.

From q + 3.6 = 11 to q = 7.4.
From \displaystyle \frac{r}{5} = 16 to r = 80.
From 12t = 84 to t = 7.
From 41 = w - 27 to 68 = w.

Writing and solving with the balance

On a balance, Mabel’s suitcase and a 3.5 kg weight on one pan balance 21 kg on the other. Write an equation and find the weight of the suitcase.
Five crates of apples, all the same weight, together balance 115 kg. Write an equation and find the weight of one crate.
The apples from one crate are shared equally into 8 baskets of 2.4 kg each. Write an equation and find the weight of the apples in the crate.
Faye’s bag weighs 16.5 kg after she takes out a 2.8 kg jacket. Write an equation and find what the bag weighed with the jacket in it.

Which moves keep the balance

The equation z + 15 = 40 has the solution z = 25. For each move, say whether a property of equality allows it, and test whether 25 still solves the new equation. (a) Add 15 to both sides. (b) Subtract 15 from the left side only. (c) Divide both sides by 5. (d) Multiply the left side by 2 and the right side by 3.
Starting from 9k = 63, Halle divides both sides by 3. Write the equation she gets, and show that k = 7 solves both equations. What move would finish the job?
Each pan of a level balance holds 27 kg. The load on each pan is doubled, then 10 kg is taken off each pan. Is the beam level, and how much is on each pan? Name the properties used.

Reasoning

Using a pan balance, explain why adding the same weight to both pans keeps the beam level, while adding weight to one pan only does not.
Multiplying both sides of an equation by 0 keeps it true. Explain why that move is never used to solve an equation.
The equation 6 \times 0 = 11 \times 0 is true. Explain what goes wrong if both sides are divided by 0.
Is “multiply both sides by \displaystyle \frac{1}{4}” an allowed move? Which other move does the same job, and why?
Explain why a move that can be undone never changes the solutions of an equation.

Answers

Subtract 17. It takes away exactly what adding 17 put on.
Multiply by 9.
26. 26 \times 11 = 286 and 286 \div 11 = 26: dividing by 11 undoes multiplying by 11.
n. Adding 35 undoes subtracting 35.
(a) and (c). In (b) the number is halved: 10 becomes 30, then 5. In (d) it ends 4 bigger.
Subtract 3.6 from both sides, by the subtraction property: 11 - 3.6 = 7.4.
Multiply both sides by 5, by the multiplication property: 16 \times 5 = 80.
Divide both sides by 12, by the division property (12 \neq 0): 84 \div 12 = 7.
Add 27 to both sides, by the addition property: 41 + 27 = 68.
s + 3.5 = 21, so s = 17.5 kg. Subtract 3.5 from both sides. Check: 17.5 + 3.5 = 21.
5c = 115, so c = 23 kg. Divide both sides by 5. Check: 5 \times 23 = 115.
\displaystyle \frac{w}{8} = 2.4, so w = 19.2 kg. Multiply both sides by 8. Check: 19.2 \div 8 = 2.4.
b - 2.8 = 16.5, so b = 19.3 kg. Add 2.8 to both sides. Check: 19.3 - 2.8 = 16.5.
(a) Allowed: z + 30 = 55, and 25 + 30 = 55. (b) Not allowed: it gives z = 40, which 25 does not solve. (c) Allowed: (z + 15) \div 5 = 8, and 40 \div 5 = 8. (d) Not allowed: it gives 2(z + 15) = 120, but 2 \times 40 = 80, so 25 does not solve it.
3k = 21. Since 9 \times 7 = 63 and 3 \times 7 = 21, k = 7 solves both. Dividing both sides by 3 again gives k = 7; dividing by 9 at the start does it in one move.
Yes, level, with 44 kg on each pan. Doubling (multiplication property) gives 54 kg each; taking 10 kg off each (subtraction property) leaves 54 - 10 = 44 kg each.
Level means both pans hold the same weight. Two equal weights that each grow by the same amount stay equal. Adding to one pan only makes it heavier, so the beam tips toward it.
It turns every equation into 0 = 0, true for every number, so it no longer picks out the solution, and it cannot be undone without dividing by 0.
It would give 6 = 11, which is false. Both 6 \times 0 and 11 \times 0 are 0, so nothing can recover which number was multiplied; division by 0 has no answer.
Yes: the multiplication property allows any number, and \displaystyle \frac{1}{4} \neq 0. It does the same job as divide both sides by 4, since a number times \displaystyle \frac{1}{4} is a quarter of it.
The move keeps a true equation true, so every solution of the first equation solves the new one. Undoing the move is also allowed, so every solution of the new equation solves the first. The solutions are exactly the same.