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# Simultaneous Equations (GCSE/14+)

Updated: Apr 6

Simultaneous equations are two (or more) equations, each with the same two (or more) unknowns and are "simultaneous" because they are solved together. The equations can only be solved fully if the number of equations is equal (or greater than) to the number of unknowns.

We can find solutions to simultaneous equations graphically and algebraically. Here we focus on the algebraic methods.

Substitution Method

The most common method to solve simultaneous equations is the substitution method. We solve the equations by substitute one unknown with an equation involving only the other unknown.

For example in these equations x=y-4 and x=6-y, we could eliminate x in the first equation by substituting the second equation into the first one. Then, we get 6-y=y-4 which is an equation with only one unknown which can be solved easily. We get y=5, which can be substitute into the first or the second equation. Finally, we get x=1.

Elimination

When we solve pair of simultaneous equations with one or more common coefficients, we can use the elimination method. We could also use the elimination method after we have multiplied one equation to create a common factor.

For example in these equations 2x+y=7 and 3x+2y=12

Since they have no common coefficients, we have to multiply the first equation by a factor of 2, and get 4x+2y=14. Now we have a pair of equations with the same coefficient of y, which is 2.

Then we can minus the second equation from the new equation:

4x+2y=14

- 3x+2y=12

x =2

And we get x =2, then we can substitute this into the first equation and get 2(2)+y=7. We solve this to get y=3.

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