U3 Linear Equations in One Unknown: Essential Formulas

General Form and Root

The standard form of a linear equation in one unknown $x$ is $ax + b = 0$, where $a$ and $b$ are real numbers and $a \neq 0$. The solution (or root) is given by the formula:

This is derived by isolating $x$: subtract $b$ from both sides to get $ax = -b$, then divide both sides by $a$.

Fundamental Properties of Equality

To solve a linear equation, we apply the following properties to both sides of the equation to maintain equality:

• Addition/Subtraction Property: If $a = b$, then $a + c = b + c$ and $a - c = b - c$.
• Multiplication/Division Property: If $a = b$ and $c \neq 0$, then $ac = bc$ and $\frac{a}{c} = \frac{b}{c}$.

The general steps are: 1) Simplify both sides (expand brackets, combine like terms). 2) Use addition/subtraction to move variable terms to one side and constant terms to the other. 3) Use multiplication/division to isolate the variable.

Key Steps for Application Problems

A common DSE question type involves forming and solving a linear equation from a word problem. Follow these steps:

1. Let the unknown quantity be $x$ (or another variable).
2. Express other related quantities in terms of $x$ using the given information.
3. Set up an equation based on the relationships described in the problem.
4. Solve the equation for $x$.
5. Check the solution against the context of the original problem.

For example: "A number increased by 7 is equal to 15." Let the number be $x$. The equation is $x + 7 = 15$. Solving gives $x = 8$.