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By the end of these notes, you will be able to:
Simultaneous equations are a pair (or group) of equations that share the same unknown variables — usually x and y. The word "simultaneous" means "at the same time," so you are looking for values of x and y that satisfy both equations at the same time.
For example:
The solution is the pair of values (x, y) that makes both statements true.
There are two main methods to solve simultaneous equations:
What it means: You manipulate the two equations so that one of the variables cancels out (is "eliminated"), leaving you with a single equation in just one unknown.
When to use it: Elimination works best when both equations are linear (no x², xy, or other powers — just plain x and y terms).
Step 1: Make the coefficient (the number in front) of one variable the same in both equations. You do this by multiplying one or both equations by a suitable number.
Step 2: Add or subtract the two equations to eliminate that variable.
Step 3: Solve the resulting single-variable equation.
Step 4: Substitute the value you found back into one of the original equations to find the second variable.
Step 5: Check your answer in both original equations.
Solve:
Step 1: Notice that the y terms are already +3y and −3y. The coefficients are the same (both 3). ✓
Step 2: Add the two equations together:
(4x + 3y) + (2x − 3y) = 1 + 14
6x = 15
Step 3: Divide both sides by 6:
x = 15 ÷ 6 = 2.5
Step 4: Substitute x = 2.5 into Equation (1):
4(2.5) + 3y = 1 10 + 3y = 1 3y = 1 − 10 = −9 y = −9 ÷ 3 = −3
Solution: x = 2.5, y = −3
Check in Equation (2): 2(2.5) − 3(−3) = 5 + 9 = 14 ✓
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