Solving Equations & Literal Equations
Every step you take must keep the two sides equal. Say which property justifies it, and the algebra never drifts.
A sealed bag and three weights sit on one pan. Seven weights sit on the other. The scale is level.
You are not allowed to open the bag. But the scale has already told you something.
Lift three weights off the left. The scale tips — so lift three off the right too, and it is level again.
Now the bag alone balances four weights. The bag holds four, and you never opened it.
Every step you took was the same step: change both pans identically, so the balance survives.
That is all "do the same to both sides" means. It is not a rule to remember — it is the only kind of move that keeps the statement true. Naming the property beside each line is just saying which move you made.
Justifying every step
Regents questions ask you to justify, not merely to solve. Name the property beside each line.
"Subtract 3x from both sides — subtraction property of equality." "Distribute — distributive property."
It feels pedantic at first, and it is exactly what stops you doing something illegal later on.
The only real rule
Do the same thing to both sides. Every legal move in algebra is a version of that one sentence.
Linear equations, reliably
Clear fractions by multiplying every term by the common denominator. Clear brackets by distributing.
Collect variables on one side, constants on the other. Then divide by the coefficient.
Check by substitution. It takes seconds and it is the only way to be certain.
Clear the fractions first
For x3 + x4 = 7, multiply everything by 12: 4x + 3x = 84, so x = 12. Much easier than adding fractions.
Literal equations
Solving A = π r2 for r uses exactly the same moves as solving a numerical equation. The letters just do not simplify.
Decide which letter you want alone, then undo everything around it in reverse order.
A = π r2 ⇒ Aπ = r2 ⇒ r = √Aπ.
Treat the other letters as numbers
In y = mx + b solved for x, the m and b behave exactly like 3 and 7 would. Subtract b, divide by m: x = y - bm.
One solution, none, or all
If the variables cancel and a false statement remains (0 = 5), there is no solution.
If a true statement remains (0 = 0), every number works and the equation is an identity.
Both outcomes are real answers. State them in words rather than leaving the page blank.
Worked examples
With fractions
Solve 2x3 - x4 = 5.
- Multiply every term by 12. 12 is the least common denominator.
- 8x - 3x = 60 12 · 2x3 = 8x and 12 · x4 = 3x.
- 5x = 60, so x = 12 Collect, then divide.
- Check: 243 - 124 = 8 - 3 = 5 ✓ Substitute into the original.
x = 12
A literal equation
Solve P = 2l + 2w for w.
- Subtract 2l: P - 2l = 2w Subtraction property of equality.
- Divide by 2: w = P - 2l2 Division property of equality.
- Or equivalently w = P2 - l. Both forms are correct; the second is often tidier.
w = P - 2l2
No solution
Solve 3(x + 4) = 3x + 10.
- Distribute: 3x + 12 = 3x + 10 Distributive property.
- Subtract 3x: 12 = 10 The variable vanishes entirely.
- 12 = 10 is false, so there is no solution. The two sides are parallel lines — never equal.
No solution
Words to know
- Literal equation
- An equation with several letters, to be rearranged for one of them.
- Properties of equality
- You may add, subtract, multiply or divide both sides by the same thing (never divide by zero).
- Extraneous solution
- A value produced by the algebra that fails in the original equation.
- Identity
- An equation true for every value, like 2(x + 3) = 2x + 6.
Mistakes to watch for
Multiplying only some terms when clearing fractions. Multiplying by the common denominator must hit every term on both sides, including the ones that were never fractions. Miss one and the equation is no longer the one you started with.
Dividing by a variable. Dividing x2 = 5x by x loses the solution x = 0, because you quietly assumed x was not zero. Factor instead and both roots survive.
Stopping at 12 = 10 without saying what it means. The variables cancelling is not a dead end — a false statement means no solution, a true one means every value works. Both are real answers worth writing down.
Losing track of which letter you are solving for. In a literal equation the other letters behave exactly like numbers. Underline the one you want alone before the first line, and undo everything around it in reverse order.
The equals sign was invented in 1557 by the Welsh mathematician Robert Recorde.
He was tired of writing "is equalle to" over and over, and chose two parallel lines because, in his words, no two things can be more equal.
His original = was much longer than ours. It took about a century to catch on; until then most books still wrote the words out.