An equation is a balance — and two simple moves keep it level.
Point 2 of 5 in this lesson: 10.1.2 The solution of an equation
Picture an old two-pan balance scale. On the left pan sits a sealed box — you cannot see inside, so you do not yet know how much it weighs. You only know that the box plus two single-pound weights balances exactly against five single-pound weights on the right. The pans hang perfectly level. From that one fact you can figure out the hidden weight without ever opening the box — and that small act of detective work is the whole idea of an equation.
In this lesson you will meet the four words that run through every equation you will ever solve: the unknown, the equation itself, its solution, and the two properties of equality that let you move toward that solution one safe step at a time. We keep a steady color habit the whole way: the unknown is violet, the add-or-subtract move is teal, the multiply-or-divide move is amber, and the solution is red. Learn to read those colors and the algebra ahead will already feel familiar.
Here is a riddle as old as arithmetic: "I'm thinking of a number. I add 4 to it, and I get 11. What number am I thinking of?" You can almost certainly answer it in your head — but watch how you answer it. There is a number you do not yet know, and there is a fact that pins it down. Algebra simply gives the unknown number a short name so we can write the fact down and work with it.
We let a letter stand in for the hidden number. By long habit that letter is x. The riddle "a number, add 4, gives 11" becomes the tidy line x + 4 = 11. The letter x is the unknown: a single number we have not found yet, but which is waiting there to be discovered.
Now be careful about two words that look alike but mean very different things. x + 4 by itself is an expression — it just names a quantity ("four more than the number"). It makes no claim; it is neither true nor false, the way the phrase "four more than my age" is neither true nor false. The moment we drop an equals sign between two expressions and claim they are the same number, we have an equation: x + 4 = 11. An equals sign is not a "compute the answer" button — it is a claim of balance, a statement that the thing on its left and the thing on its right are one and the same number.
| Expression | Equation |
|---|---|
| names a quantity | claims two quantities are equal |
| no equals sign | has exactly one equals sign |
| x + 4 | x + 4 = 11 |
| cannot be true or false | true for the right x, false for the wrong one |
An expression names a number; an equation sets two expressions equal with an = sign and claims they balance. The letter x is the unknown — one number we have not found yet.
An equation is a claim, and a claim invites a question: for which number does the claim come true? A solution of an equation is a value that, when you drop it in for x, makes the two sides truly equal — it makes the balance hang level. Finding that value is what we mean by "solving" the equation.
You do not need a method yet to check whether a particular number is the solution; you just substitute it and see whether both sides come out the same. This habit — checking by substitution — is your safety net for every equation you will ever solve.
Test whether x = 7 solves x + 4 = 11. Put 7 in for x: the left side becomes 7 + 4 = 11, and the right side is 11. The two sides match, 11 = 11 ✓ — so x = 7 is the solution.
Now test x = 5. The left side becomes 5 + 4 = 9, but the right side is 11. Since 9 ≠ 11 ✗, the scale would tilt — so x = 5 is not a solution. A wrong guess is not "close enough"; it simply fails the balance test.
So the solution is the one value of x that keeps both pans level. Try other numbers and the scale tips: too small a box and the left pan rises, too large and it sinks. There is exactly one box-weight that makes x + 4 sit even with 11, and that weight is 7.
Step the hidden weight x up or down. Watch the beam tilt, and stop when the scale reads "balanced." That value of x is the solution.
A solution is a value of x that makes the equation true. Always check by substitution: put your answer back in and confirm the two sides are equal.
Guessing values works for tiny puzzles, but we want a method that discovers the answer instead of hunting for it. The balance scale hands us the first tool. Imagine the scale hanging level. If you place the same weight on both pans, it is still level. If you lift the same weight off both pans, it is still level. The scale does not care what the weights are — as long as you treat both sides exactly alike, the balance is preserved.
In symbols, this is the first basic property of equality:
if a = b, then a + c = b + c and a − c = b − c
You may add the same amount to both sides of an equation, or subtract the same amount from both sides, and the new equation has exactly the same solution as the old one. This is the move we color teal all through Stage 10.
Watch it do real work on x + 4 = 11. The box is buried under 4 extra unit weights on the left. To uncover it, lift those 4 weights off the left pan — but to keep the scale honest, lift 4 weights off the right pan at the very same time. Left: x + 4 − 4 leaves just x. Right: 11 − 4 leaves 7. The box now sits alone against 7 weights, and we can read the answer straight off the scale: x = 7.
| x + 4 = 11 | the equation |
| x + 4 − 4 = 11 − 4 | − 4 from both sides |
| x = 7 | the box stands alone |
Press the button to lift one unit weight from each pan at once. The scale stays level the whole time — until the box marked x stands alone and the answer appears.
Add or subtract the same amount on both sides and the equation keeps its solution. This is how we peel a number off the side that holds x.
Adding and subtracting handle a number stuck beside x. But what about a number multiplying x, as in 3x = 12? Here the left pan does not hold "x and some extra weights" — it holds three identical boxes, each weighing x, balancing 12 weights. Lifting a fixed number of weights will not free a single box. We need a different, scaling move.
Back to the balance. If a level scale is fair, then making both pans three times as heavy keeps it level, and so does keeping just a third of each pan. Scale both sides by the same factor and they still match. In symbols, this is the second basic property of equality:
if a = b, then a · c = b · c and a ÷ c = b ÷ c (provided c ≠ 0)
You may multiply both sides by the same number, or divide both sides by the same nonzero number, and keep the same solution. This is the move we color amber. Use it on 3x = 12: three equal boxes balance 12 weights, so one box must balance a third of 12. Divide both pans by 3 — share 12 weights fairly among the 3 boxes, 4 each — and one box stands against 4 weights: x = 4. Check: 3 · 4 = 12 ✓.
| 3x = 12 | the equation |
| 3x ÷ 3 = 12 ÷ 3 | ÷ 3 on both sides |
| x = 4 | one box against 4 weights |
The second property comes with one fence around it: the number you scale by must be nonzero. Multiplying both sides by 0 turns any equation into the empty truth 0 = 0 — true no matter what x is, so you have thrown away every clue about x. And dividing by 0 is simply undefined; it is not a number you are allowed to share things into 0 of. So you may scale by 2, by −5, by 13 — by anything at all except 0.
Choose a factor and apply it to both pans. Multiplying or dividing fairly keeps the scale level; the value of x never changes — only how the equation looks.
Multiply or divide both sides by the same nonzero number and the equation keeps its solution. Dividing by the coefficient is how we turn "several boxes" into "one box."
Now the two properties join forces. Together they let you peel everything off x, one careful move at a time, until the box stands utterly alone and the scale reads the answer. Real equations stack both kinds of clutter at once: take 2x + 3 = 11, where x is both doubled and buried under 3 extra weights.
The order is the natural one: undo the adding first, then undo the multiplying. First reach for the teal move and subtract 3 from both sides to clear the loose weights; that leaves 2x = 8. Then reach for the amber move and divide both sides by 2 to split the two boxes evenly; that leaves x = 4. Notice each line is still a true balance — we only ever did the same thing to both sides.
| 2x + 3 = 11 | the equation |
| 2x + 3 − 3 = 11 − 3 | − 3 from both sides |
| 2x = 8 | the loose weights are gone |
| 2x ÷ 2 = 8 ÷ 2 | ÷ 2 on both sides |
| x = 4 | the box stands alone |
Never skip the last step — the check. Put 4 back into the original equation: 2 · 4 + 3 = 8 + 3 = 11, and the right side is 11. So 11 = 11 ✓, and we are certain x = 4 is the solution. The check is not busywork; it catches a slipped sign or a dropped term every time.
Do the same to both sides, in order: first subtract 3, then divide by 2. Watch the box marched toward standing alone, and the scale stay level the whole way.
Stack the two properties to isolate x: clear the added numbers first (teal), then clear the multiplier (amber), then check. In Lesson 10.2 we turn this into one tidy routine that solves any linear equation in one unknown.
A letter like x stands for an unknown number; an expression names a quantity, but an equation puts an equals sign between two expressions and claims they balance. A solution is the value of x that makes that claim true — always confirm it by substitution. To find it, lean on the balance and its two properties of equality: add or subtract the same amount on both sides, and multiply or divide both sides by the same nonzero number. Stack those moves to peel everything off x until it stands alone, and you have the solution.
You now have the two legal moves and the instinct for the balance. Next we forge them into a single dependable routine — a step-by-step recipe that solves every linear equation in one unknown, even ones with the unknown on both sides or fractions in the way. See Lesson 10.2 — Solving Linear Equations in One Unknown.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson lays the conceptual foundation for solving equations and serves several Common Core standards: 6.EE.B.5 (a solution is a value that makes an equation true, tested by substitution), 6.EE.B.6 (a letter stands for an unknown number), and 6.EE.B.7 (solve equations of the forms x + p = q and px = q). The two properties developed here — the addition and multiplication properties of equality — are exactly the principles that underlie the formal equation-solving of 8.EE.C.7. Pedagogically we hold the balance-scale model in front of every move, so that "do the same to both sides" is something the child sees rather than memorizes.
The #1 misconception to watch for is reading the equals sign as a one-way "compute the answer" button (the calculator meaning) rather than as a symmetric claim of balance. A child with this misread will happily write things like 3 + 4 = 7 + 2 = 9, or will resist "doing the same to both sides" because only one side seems to be "the problem." The antidote is the scale itself: an equals sign means the two pans hang level, the relationship is symmetric (if a = b then b = a), and any legal move must touch both pans equally. Keep returning to "is it still level?" and the correct meaning takes hold.