Detailed Summary
The elimination method is a powerful algebraic technique used for solving pairs of linear equations. It aims to eliminate one variable by manipulating the equations through addition or subtraction after ensuring equal coefficients for that variable. The standard steps for applying the elimination method include:
- Adjusting Coefficients: Each equation is multiplied by appropriate constants, if necessary, to make the coefficients of a chosen variable (either x or y) equal.
- Eliminating Variables: By adding or subtracting the equations, one variable is eliminated, resulting in a simpler equation in one variable.
- Solving the Remaining Equation: The equation is solved for the remaining variable, which is then substituted back into one of the original equations to find the other variable.
- Identifying Types of Solutions: If a true statement results (like 0 = 0), the system is dependent with infinite solutions. If a false statement appears (like 0 = 9), the system is inconsistent with no solution.
For illustration, we solve real-world problems, such as calculating incomes based on savings, or finding two-digit numbers whose digits differ, applying the method through multiple examples. The significance of the elimination method lies in its ability to simplify complex systems into manageable equations, making it easier to visualize and solve.
Example:
Use the elimination method to find all possible solutions of the following pair of linear equations:
$$
3x + 5y = 11 \quad (1)
5x + 7y = 19 \quad (2)
$$
Solution:
Step 1: Multiply Equation (1) by 5 and Equation (2) by 3 to make the coefficients of \(x\) equal. Then we get the equations as:
$$
15x + 25y = 55 \quad (3)
15x + 21y = 57 \quad (4)
$$
Step 2: Subtracting Equation (4) from Equation (3), we have:
$$
(15x + 25y) - (15x + 21y) = 55 - 57
$$
$$
4y = -2
$$
Thus,
$$
y = -\frac{1}{2}, ext{ which produces unique solutions.}
$$
Therefore, the pair of equations has a unique solution.