Linear Equations — Two Variables
- Formula / Rule
- Linear Equations — Two Variables
- Topic
- Algebra
- Formula
Given: $a_1x + b_1y = c_1$ $a_2x + b_2y = c_2$
Solving Methods:
- Substitution method: Express one variable in terms of the other, then substitute into the second equation.
- Elimination method: Multiply equations to make coefficients of one variable equal, then add or subtract them.
Conditions for Solutions:
- Unique solution: $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$
- Infinite solutions: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
- No solution: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
- When to Use
- Problems involving two unknowns — age, mixture, pricing, etc.
- Worked Example
Q: $x + y = 10$, $x - y = 4$. Find $x$ and $y$.
Adding equations: $2x = 14 \rightarrow \textbf{x = 7}$ Substituting $x$: $y = 10 - 7 = \textbf{3}$