Solving simultaneous equations involves finding the values of the unknown variables that satisfy all equations in the system. The most common method to achieve this is the elimination method.
Understanding the Elimination Method
The elimination method focuses on removing one of the unknowns from each equation to simplify the problem. This makes it possible to solve for the remaining unknown. Here's how it works:
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Identify the Unknown to Eliminate: Look at the coefficients (the number multiplied by the variable) of the unknowns in both equations.
- For example, in the equation 3a + b = 7, the coefficient of a is 3.
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Manipulate Equations (if needed): Multiply one or both equations by a constant so that the coefficients of the unknown you want to eliminate are either the same or the negative of each other.
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Eliminate the Unknown: Add or subtract the equations to eliminate the chosen unknown. If the coefficients are the same, subtract the equations. If the coefficients are the negative of each other, add the equations.
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Solve for the Remaining Unknown: After eliminating one unknown, you'll have a single equation with one unknown. Solve this equation to find the value of that unknown.
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Substitute to Find the Other Unknown: Substitute the value you just found back into one of the original equations (or any equation with both unknowns) and solve for the remaining unknown.
Example
Let's say you have these two equations:
- 2x + y = 7
- x - y = 2
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Notice that the coefficients of 'y' are +1 and -1. This is ideal for elimination by adding the equations.
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Add equation 1 and equation 2:
(2x + y) + (x - y) = 7 + 2
3x = 9 -
Solve for x:
x = 9 / 3
x = 3 -
Substitute x = 3 into equation 2:
3 - y = 2 -
Solve for y:
-y = 2 - 3
-y = -1
y = 1
Therefore, the solution is x = 3 and y = 1.
Summary
The elimination method involves strategically manipulating and combining equations to eliminate one variable at a time until a solution can be found for all variables.
