The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 2 (Intermediate Algebra) Solving Linear Equations

Overview

This lecture is a thorough tour of solving linear equations — equations in which the variable appears only to the first power. We start from the simplest one-step cases and build up systematically: subtracting a constant, dividing by a coefficient, distributing, combining like terms, collecting variables onto one side, and clearing fractions. The unifying idea is isolating the variable: whatever you do to one side of an equation, you must do to the other, until the variable stands alone. The final segment generalizes the same moves to literal equations, where you solve for one letter in terms of others (e.g. rearranging a formula). Mastering this routine is the foundation for nearly everything that follows in algebra.

Key Concepts

What makes an equation linear

A linear equation is one in which the variable is raised to a single power, $x^1$ (i.e. just $x$ ).
Not linear: $ax^2 + b = c$ (squared term), $\sqrt{x} = c$ (since $\sqrt{x} = x^{1/2}$ ), an $x$ in a denominator ( $x^{-1}$ ), or $\log x$ .

The two inverse operations

Undo addition with subtraction: to remove a term added to the variable, subtract it from both sides.
Undo multiplication with division: to remove a coefficient, divide both sides by it — equivalently, multiply both sides by its reciprocal. Thinking "multiply by $\tfrac{1}{a}$ " rather than "divide by $a$ " helps keep signs and fractions straight.

General form and recipe

For $ax + b = c$ (with $a,b,c$ any numbers): subtract $b$ , then multiply by $\tfrac{1}{a}$ : $x = \frac{c - b}{a}$

Sign rule for products

An even number of negative factors gives a positive result; an odd number gives a negative. Example: $(-1)^{47} = -1$ , while $(-2)^6 = 2^6 = 64$ (positive, even count).

Order of operations when solving

Distribute to clear parentheses: $a(x - y) = ax - ay$ .
Combine like terms on each side.
Collect all variable terms on one side and all constants on the other (usually move the variable to the side that already has more of it, to avoid a negative coefficient).
Isolate the variable by dividing by its coefficient.

Handling a negative variable term

$-x$ means $-1 \cdot x$ , so dividing both sides by $-1$ flips the sign. If a leading $-ax$ is awkward, you may add $ax$ to both sides first to make the coefficient positive (more steps, but optional).

Fractions don't change the method

Treat fractional coefficients exactly like whole numbers: subtract the constant, then multiply by the reciprocal of the coefficient. Combine fractions over a common denominator as needed. To rewrite a fraction with a new denominator, multiply it by a form of $1$ (e.g. $\tfrac{3}{3}$ ).

Literal equations (solving for a variable)

The same isolating moves work when other letters stand in for numbers. To solve for a chosen variable, get it alone on one side.
If a target variable is trapped inside a product like $a(x-y) = b$ and you want $a$ , divide by the whole factor $(x-y)$ — do not distribute. If instead you want $x$ , you must distribute first.

Worked Example

Solve a multi-step linear equation with variables on both sides:

\begin{aligned} 4(2x - 6) + 9 &= 3(x - 7) + 8x \\ 8x - 24 + 9 &= 3x - 21 + 8x \\ 8x - 15 &= 11x - 21 \\ 8x - 15 - 8x &= 11x - 21 - 8x \\ -15 &= 3x - 21 \\ -15 + 21 &= 3x \\ 6 &= 3x \\ x &= 2 \end{aligned}

Check by substituting $x = 2$ : the left side gives $4(4-6)+9 = 4(-2)+9 = 1$ , and the right side gives $3(2-7)+16 = 3(-5)+16 = 1$ . Both sides agree, so $x = 2$ is correct.

For a literal-equation example, solving $A = \pi r^2 + \pi r s$ for $s$ :

\begin{aligned} A - \pi r^2 &= \pi r s \\ s &= \frac{A - \pi r^2}{\pi r} \end{aligned}

Summary

A linear equation has the variable to the first power only; the goal of solving is always to isolate that variable.
Use inverse operations — subtract to undo addition, divide (multiply by the reciprocal) to undo a coefficient — and apply every step to both sides.
Work in order: distribute, combine like terms, collect variables on one side and constants on the other, then divide by the coefficient.
The general result $ax + b = c \Rightarrow x = \dfrac{c-b}{a}$ captures the whole pattern.
Fractions use the identical method; just find a common denominator and multiply by the reciprocal of the fractional coefficient.
Literal equations are solved the same way — divide by an entire factor to free a variable, and always check a solution by substituting it back in.