Example of a Equation in Math: Worked Examples by Type

The fastest way to learn equations is to watch them get solved - so this page is a gallery, not a lecture. Picture a pan balance: the left pan holds $x+7$, the right pan holds $15$, and the beam sits level. That level beam is the whole secret behind every example of a equation in math you will meet.

Each solved sample below tips the beam with one legal move, then levels it again. Read a few, borrow the pattern, and the moves start to feel obvious. The gallery runs from one-step linear equations up to quadratics, and every example of a equation in math along the way keeps the beam level.

Every example of a equation in math runs on one rule: whatever you do to one side, do to the other. Ray's 1866 Higher Algebra turns that into a four-step rule for a simple equation. Clear fractions first. Transpose the unknown terms to one side, the known terms to the other. Reduce each member, then divide by the coefficient of the unknown.

Ray also names the habit this page keeps in every example of a equation in math. Substitute the answer into the original equation. If the two members come out equal, the value is true. He calls that verification - One unknown, two members, legal moves on both sides, a check. That skeleton never changes - only the costume does: parentheses, fractions, a second unknown, or a square.

This gallery is a sample of equation families, and each example of a equation in math is placed with its own kind, easiest first:

• One-variable linear - Examples 1-3: one $x$, moves get longer.
• With parentheses - Examples 4-5: expand first, then solve.
• With fractions - Examples 6-7: clear denominators, then solve.
• Two unknowns - Examples 8-9: a pair of equations, solved together.
• Quadratic - Examples 10-11: a square enters, two answers leave.

Mathematical equations come in families, and each family has one signature move. Sample math equations from every family sit below, solved in two or three steps each, so you can compare the moves side by side. If you only need one type, jump straight to its pair.

The gentlest example of a equation in math is one move wide.

Equation. $x + 7 = 15$

Solve. Seven is added to $x$, so subtract $7$ from both sides: $x = 15 - 7 = 8$.

Check. $8 + 7 = 15$ - level beam, so $x = 8$.

One more term joins, so this example of a equation in math takes two moves - undo them in reverse order.

Equation. $2x + 5 = 13$

Solve. Subtract $5$ from both sides: $2x = 8$. Divide both sides by $2$: $x = 4$.

Check. $2 \times 4 + 5 = 13$ - true, so $x = 4$.

Real sample math equations often put $x$ on both pans. This example of a equation in math collects them first.

Equation. $7x = 3x + 12$

Solve. Subtract $3x$ from both sides: $4x = 12$. Divide by $4$: $x = 3$.

Check. Left: $7 \times 3 = 21$; right: $3 \times 3 + 12 = 21$. Equal, so $x = 3$.

Every example equation above is linear with no brackets and no fractions - the plainest family.

Ray's exercise book supplies this example of a equation in math with brackets - expand before you move anything.

Equation. $5(x+1) - 2 = 3(x+5)$

Solve. Expand: $5x + 5 - 2 = 3x + 15$, that is $5x + 3 = 3x + 15$. Subtract $3x$: $2x + 3 = 15$. Subtract $3$ and divide by $2$: $x = 6$.

Check. Left: $5 \times 7 - 2 = 33$; right: $3 \times 11 = 33$. So $x = 6$.

A second Ray exercise, and this example equation hides a negative expansion - watch the signs.

Equation. $3(x-2) + 4 = 4(3-x)$

Solve. Expand: $3x - 6 + 4 = 12 - 4x$, that is $3x - 2 = 12 - 4x$. Add $4x$: $7x - 2 = 12$. Add $2$ and divide by $7$: $x = 2$.

Check. Left: $3 \times 0 + 4 = 4$; right: $4 \times 1 = 4$. So $x = 2$.

Fractions look scary until one move kills them - a fraction example of a equation in math always starts by clearing denominators, the way Ray does it.

Equation. $\frac{x}{3} - \frac{x}{4} = 1$

Solve. Multiply every term by $12$, the L.C.M. of the denominators: $4x - 3x = 12$. So $x = 12$.

Check. $\frac{12}{3} - \frac{12}{4} = 4 - 3 = 1$ - true, so $x = 12$.

Ray's worked example of a equation in math mixes a fraction with a bracketed numerator - his steps, his numbers.

Equation. $3x - \frac{24-2x}{7} = x + 8$

Solve. Clearing of fractions: $21x - (24 - 2x) = 7x + 56$, so $21x - 24 + 2x = 7x + 56$. Transposing: $16x = 80$. Dividing: $x = 5$.

Check. Ray verifies in the original: $3 \times 5 - \frac{24-10}{7} = 15 - 2 = 13$, and $5 + 8 = 13$; so $13 = 13$ - true.

A sample of equation pairs: with two unknowns, one example of a equation in math is not enough information - you need two equations.

Equations. $x + y = 12$ and $2x - y = 3$

Solve. Add the two equations - the $y$ terms cancel: $3x = 15$, so $x = 5$. Back-substitute: $5 + y = 12$, so $y = 7$.

Check. $5 + 7 = 12$ and $2 \times 5 - 7 = 3$ - both level, so $(x, y) = (5, 7)$.

Same family, second sample math equations style - another example of a equation in math with two unknowns. Solve one equation for one letter, then swap it in.

Equations. $3x + 2y = 16$ and $x - y = 2$

Solve. From the second, $x = y + 2$. Substitute into the first: $3(y+2) + 2y = 16$, so $5y = 10$ and $y = 2$. Then $x = 4$.

Check. $3 \times 4 + 2 \times 2 = 16$ and $4 - 2 = 2$ - so $(x, y) = (4, 2)$.

Ray's first pure quadratic is the richest example of a equation in math on this page - and the first with two answers.

Equation. $\frac{1}{3}x^2 - 3 + \frac{5}{12}x^2 = \frac{51}{4} - x^2$

Solve. Clearing of fractions ($\times 12$): $4x^2 - 36 + 5x^2 = 153 - 12x^2$. Transposing and reducing: $21x^2 = 189$, so $x^2 = 9$. Extracting the square root: $x = \pm 3$.

Check. At $x = 3$: left $3 - 3 + \frac{15}{4} = \frac{15}{4}$; right $\frac{51}{4} - 9 = \frac{15}{4}$. Since $(-3)^2 = 3^2$, the value $x = -3$ works identically - Ray says so, and the squares agree.

The last example equation in the quadratic family starts from a square bracket.

Equation. $(x+2)^2 = 4x + 5$

Solve. Expand: $x^2 + 4x + 4 = 4x + 5$. The $4x$ terms cancel: $x^2 = 1$, so $x = \pm 1$.

Check. At $x = 1$: $9 = 9$; at $x = -1$: $1 = 1$ - both true.

A cousin pattern: $(x-4)(x+1) = 0$ is already factored, and a product is zero only when a factor is zero, so $x = 4$ or $x = -1$. Two shapes, same two-answer signature that separates quadratics from every example of a equation in math earlier in the gallery.

1. Moving terms without the other side. Writing $x + 7 = 15$ as $x = 15$ forgets the $7$; the beam tips. Fix: every move happens on both sides, no exceptions.

2. Dropping signs before a minus fraction. Ray's remark on his own example: when a fraction preceded by a minus sign is cleared, the signs of all terms in the numerator must change - $21x - (24 - 2x)$ becomes $21x - 24 + 2x$, not $21x - 24 - 2x$.

3. Stopping at one root of a quadratic. $x^2 = 9$ gives $x = 3$ and $x = -3$; writing only $3$ loses half the answer. Every example of a equation in math with a square owes two checks.

4. Checking in the transformed equation. In a long example of a equation in math, Ray substitutes into the original. A slip made while transforming would hide itself inside the transformed copy.