Mathematics from zero
Equations
Picture a balance scale, level and still. On the left pan: a mystery weight plus a 3-gram block. On the right pan: a 7-gram block. The pans balance — so the mystery weight must be exactly 4 grams. You just solved an equation.
After this lesson you can say what an equation is, explain what solving one means, use the balance rule to keep an equation true, and solve a simple equation by undoing its operations with inverses.
An equation says two expressions are equal. Unlike a bare expression, an
equation has an equals sign, and it makes a claim: the thing on the left and the
thing on the right are the same number. x + 3 = 7 claims that x + 3 and 7 are
equal. The equation is a balanced scale, and = is the level point.
Solving an equation means finding the value of the variable that makes it true. For
x + 3 = 7, the solution is the number you can put in place of x so both sides
really are equal. Here it is x = 4, because 4 + 3 truly equals 7. Solving is the
search for that value.
The balance rule: whatever you do to one side, do to the other. A scale stays level only if both pans change together. The same is true of an equation: if you add, subtract, multiply, or divide on one side, you must do the identical thing to the other side. Then the two sides stay equal, and the solution is unchanged.
Solve by undoing operations with their inverses. To get the variable alone, peel
away whatever is attached to it — using the opposite operation. x + 3 = 7 has a + 3
on the variable; undo it by subtracting 3 from both sides: x = 4. Addition is
undone by subtraction, multiplication by division. Keep undoing until the variable
stands by itself.
Solve 2x + 1 = 9.
The variable x has two things attached: it is multiplied by 2, then 1 is added. Undo
them in reverse order, outermost first.
Undo the + 1: subtract 1 from both sides. Left becomes 2x, right becomes 8. Now
the equation is 2x = 8.
Undo the × 2: divide both sides by 2. Left becomes x, right becomes 4. So
x = 4.
Check by substituting back into the original: 2 × 4 + 1 = 8 + 1 = 9. It matches, so
x = 4 is correct.
Why this works
Why must you do the same thing to both sides? Because the equation’s whole claim is that the two sides are equal. Change one side only and that claim breaks — you would be describing a different equation with a different solution. Doing the identical move to both sides keeps the scale level, so the value of the variable never shifts.
Common mistake
The most common mistake is operating on one side only — subtracting 1 from the left of
2x + 1 = 9 but leaving the right as 9. That silently changes the equation. Every
move must touch both sides, identically. If you subtract 1, subtract it left and
right.
Solve x + 5 = 12. Type the value of x.
Solve x − 3 = 10. Type the value of x.
Solve 2x = 14. Type the value of x.
Solve 3x + 2 = 20. Type the value of x.
Solve x ÷ 2 = 5. Type the value of x.
To solve 2x + 1 = 9 you subtract 1. Where must the subtraction happen?
An equation says two expressions are equal, marked by the equals sign. Solving it means finding the variable’s value that makes the claim true. The balance rule keeps the equation honest: whatever you do to one side, do to the other. Solve by undoing the operations attached to the variable with their inverses — subtraction undoes addition, division undoes multiplication — until the variable stands alone.