2.2. Elimination Method

Used to eliminate one variable by adding or subtracting the equations.

The elimination method is a technique used to solve simultaneous equations by removing one variable so that the equations can be simplified. This is particularly useful when dealing with equations that have similar coefficients for one of the variables. The goal is to manipulate the equations in such a way that the variable you want to eliminate has the same coefficient in both equations, making it easy to add or subtract the equations.

Imagine you have two bags of apples and oranges, and you want to find out how many apples and oranges you have in total. By adding or subtracting the counts of apples and oranges in various combinations, you can determine the amounts without directly asking for each separately.

Steps:

1. Multiply one or both equations (if necessary) so coefficients of one variable match.
2. Add or subtract equations to eliminate one variable.
3. Solve the resulting equation.
4. Substitute to find the other variable.

The elimination method consists of a series of clear steps. First, you may need to multiply one or both equations to align the coefficients of one variable, making them the same. Once the coefficients are aligned, you can either add or subtract the equations to eliminate that variable from consideration. With one variable removed, you're left with a single equation that can be easily solved. After finding the solution for one variable, you substitute it back into one of the original equations to find the value of the second variable.

Think of this step-by-step process like cooking a dish. First, you gather your ingredients (multiply the equations), then mix them together (add or subtract), cook them on the stove (solve the equation), and finally plate your meal (substitute to find the other variable). Each step is necessary to end up with the final dish, just like you need to complete each part of the method to find the solution to the equations.

Example: Solve: 2𝑥 + 3𝑦 = 12 4𝑥 − 3𝑦 = 6 Solution: Add equations: (2𝑥 + 3𝑦) + (4𝑥 − 3𝑦) = 12 + 6 ⇒ 6𝑥 = 18 ⇒ 𝑥 = 3 Substitute 𝑥 = 3 into first equation: 2(3) + 3𝑦 = 12 ⇒ 6 + 3𝑦 = 12 ⇒ 𝑦 = 2 Solution: 𝑥 = 3, 𝑦 = 2

In this example, we have two equations: 2𝑥 + 3𝑦 = 12 and 4𝑥 − 3𝑦 = 6. To use the elimination method, we can add these two equations directly. Notice that the terms +3𝑦 and -3𝑦 cancel each other out when added together. This gives us a new equation 6𝑥 = 18. Solving for 𝑥 gives us 𝑥 = 3. Next, we substitute this value back into one of the original equations, specifically the first one. By replacing 𝑥 with 3, we can solve for 𝑦, leading us to the final answer of 𝑦 = 2. Hence, the solution to the simultaneous equations is 𝑥 = 3 and 𝑦 = 2.

Consider two people, Alex and Jamie, planning a fruit stall. Alex has a total of 12 pieces of fruit made up of apples and oranges, while Jamie has a total of 6 pieces of fruit, all in relation to their sales. Using the elimination method, Alex combines their counts to find out how many apples and oranges they have without necessarily counting each fruit type separately. By strategically eliminating one fruit type, they can clearly see the counts they need.

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