Steps for Elimination Method
The elimination method eliminates one variable by making its coefficients equal in both equations:
Multiply one or both equations by suitable non-zero constants so that coefficients of one variable become numerically equal.
Add or subtract the equations to eliminate that variable, giving an equation in one variable.
Solve for the remaining variable, then substitute back into any original equation to find the second variable.
Step-by-Step Problem Solving
Solve 9x - 4y = 2000 and 7x - 3y = 2000 using Elimination
Step 1: Multiply to make y coefficients equal:
Multiply Eq(1) by 3: 27x - 12y = 6000 — (3)
Multiply Eq(2) by 4: 28x - 12y = 8000 — (4)
Step 2: Subtract Eq(3) from Eq(4) to eliminate y:
x = 2000
Step 3: Substitute x = 2000 in Eq(1):
18000 - 4y = 2000 ⟹ 4y = 16000
y = 4000
Solution: x = 2000 and y = 4000.
Sum of digits problem
Problem: The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits differ by 2, find the number.
Let tens digit = x and units digit = y. Number = 10x + y.
Reversed number = 10y + x.
11x + 11y = 66 ⟹ x + y = 6 — (1)
If x - y = 2, adding (1) and (2) gives 2x = 8 ⟹ x = 4, y = 2.
The numbers are 42 or 24.
Key Takeaways
If signs of the variable to be eliminated are opposite, ADD the equations.
If signs of the variable to be eliminated are same, SUBTRACT the equations.
Elimination is usually faster than substitution for equations with integer coefficients!