6.1 Linear Equations in One Variable
1. Linear Algebraic Terms and Expressions
A linear term in one variable is a term where the variable has a power (exponent) of 1.
- Examples of linear terms: $3x$, $-5y$, $\frac{z}{2}$.
- Non-linear terms: $x^2$, $y^3$, $\sqrt{x}$, $\frac{1}{x} = x^{-1}$.
2. Linear Equations in One Variable
A linear equation in one variable is an equality statement ($=$) containing only one variable, where the highest power of that variable is 1.
- General Form: $ax + b = 0$, where $a \neq 0$ and $a, b$ are constants.
- Examples: $2x + 5 = 11$, $3 - y = 8y$.
3. Solving Linear Equations in One Variable
Solving an equation means finding the numerical value of the variable that makes the equation true (the root or solution).
- Methods:
- Trial and Improvement Method: Substituting estimated values until both sides are equal.
- Backtracking / Inverse Operations Method: Undoing operations step-by-step.
- Equality Properties Method: Performing the same arithmetic operation on both sides of the equal sign.
- Example: Solve $3x - 4 = 11$:
3x - 4 + 4 = 11 + 4
3x = 15 &implies; x = 15 / 3 = 5
6.2 Linear Equations in Two Variables
1. Concept of Linear Equations in Two Variables
A linear equation in two variables contains two distinct variables, each raised to the power of 1, with no product of variables (like $xy$).
- General Form: $ax + by = c$, where $a \neq 0, b \neq 0$.
- Examples: $2x + 3y = 12$, $p - q = 5$.
- Solutions: A linear equation in two variables has infinitely many ordered pairs $(x, y)$ as solutions.
6.3 Simultaneous Linear Equations in Two Variables
1. Concept of Simultaneous Linear Equations
Two linear equations involving the same two variables processed together form a set of simultaneous linear equations. The solution is an ordered pair $(x, y)$ that satisfies both equations at the same time.
2. Methods to Solve Simultaneous Equations
- Graphical Method: Plot both equations on a Cartesian plane. The point of intersection $(x, y)$ is the solution.
- Substitution Method: Express one variable in terms of the other from one equation, then substitute it into the second equation.
- Elimination Method: Add or subtract the equations to eliminate one variable when its coefficients are equal or opposite.
Example (Elimination): Solve $x + y = 7$ and $x - y = 1$:
- Add the two equations: $(x + y) + (x - y) = 7 + 1 \implies 2x = 8 \implies x = 4$.
- Substitute $x = 4$ into $x + y = 7 \implies 4 + y = 7 \implies y = 3$.
- Solution: $x = 4, y = 3$.