Solving Linear Equations

What a linear equation is

An equation is a statement that two expressions are equal, written with an equals sign between them, such as $3x - 5 = 7$. A solution of an equation is a value of the variable that makes that statement true. You already have the tool for testing a candidate, namely substitution: replace the variable with the value and check whether the two sides come out equal.

Is $x = 4$ a solution of $3x - 5 = 7$? Substitute and compute the left side:

The left side equals $7$, the same as the right, so $x = 4$ is a solution. The value $x = 2$ is not, because $3(2) - 5 = 1$, which is not $7$. Solving an equation means finding every value that passes this test.

This lesson is about linear equations in one variable: equations in which the variable appears only to the first power. It is never squared, never under a root, never in a denominator, so $x^2 = 9$, $\sqrt{x} = 3$, and $\frac{6}{x} = 2$ all fall outside our scope for now. The linear equations you will solve in this lesson can be rearranged into the standard form

where $a$, $b$, and $c$ are numbers and $a \neq 0$, which gives exactly one solution. A few one-variable equations instead collapse to the special case $a = 0$, with either no solution or infinitely many, and the next lesson handles those. The name linear comes from the straight line such an equation graphs, a picture the graphing chapter will draw later. Here the point is that the variable stands alone to the first power, which makes these the most basic equations to solve and the foundation for everything after.

Keeping the balance: the properties of equality

Picture an equation as a balance scale. The equals sign is the pivot, and the two expressions are equal weights in the two pans, holding the beam level. This picture is the key to solving, because it tells you exactly which moves are legal. Anything you do to one pan alone tips the scale and breaks the equation. But if you do the same thing to both pans, the beam stays level and the equality survives.

Four such moves, the properties of equality, are the only tools you need. If $a = b$, then for any number $c$:

• $a + c = b + c$ (add the same amount to both sides),
• $a - c = b - c$ (subtract the same amount from both sides),
• $a \cdot c = b \cdot c$ (multiply both sides by the same nonzero number),
• $\dfrac{a}{c} = \dfrac{b}{c}$ (divide both sides by the same nonzero number).

Each move produces a new equation, and the new equation has exactly the same solutions as the old one. Two equations with the same solution set are called equivalent. Solving works by walking through a chain of equivalent equations, each simpler than the last, until the variable stands alone. Here is why the moves preserve the solutions.

Isolating the variable

To solve is to transform the equation, one legal move at a time, into the form $x = (\text{a number})$. The strategy is to undo the operations wrapped around the variable, in reverse order. Building an expression like $2x + 3$ from $x$ means first multiplying by $2$ and then adding $3$. To take that expression apart, you reverse both the operations and their order, subtracting $3$ first and dividing by $2$ last. Each undoing uses an inverse operation: addition and subtraction undo each other, and multiplication and division undo each other.

Applied to the standard form, this reverse-order idea always lands on a single value.

Clearing parentheses and combining like terms

Most equations do not arrive in tidy standard form. They come with parentheses and with several like terms scattered on a side. So before you isolate the variable, you first simplify each side on its own, using the skills from the last two lessons. Distribute to clear any parentheses, then combine like terms, and only then start undoing operations. Keep the two jobs separate: finish simplifying a side before you move any term across the equals sign.

Variables on both sides

So far the variable has lived on one side. When it appears on both, as in $5x + 2 = 3x + 10$, you cannot isolate it until it occupies a single side, and the subtraction property is what gets it there. Subtract the smaller variable term, $3x$, from both sides. On the right, $3x - 3x = 0$, so the variable leaves that side entirely; on the left, $5x - 3x = 2x$:

What remains is an ordinary two-step equation. Subtracting the smaller coefficient, $3x$ rather than $5x$, is a deliberate choice. That choice leaves the variable with a positive coefficient, which heads off the sign slips that a negative coefficient invites. From there you solve as before, and the balance never tipped, because every term you removed left from both pans at once.

Putting the steps together

Every linear equation, however tangled, yields to the same ordered routine. Nothing in it is new; it just fixes the order in which to spend the tools you already have.

1. Clear parentheses by distributing across each one.
2. Combine like terms on each side separately.
3. Collect the variable on one side and the constants on the other, using the addition and subtraction properties.
4. Divide by the coefficient of the variable to leave it alone.
5. Check by substituting the answer into the original equation.

The last step is not optional. A single arithmetic slip anywhere above produces a wrong value, and substituting back into the original equation is how you catch it. That check works because the original is the one line you know for certain was written correctly.

Common mistakes