10. Solve using the addition method

If we triple the second equation, the coefficients on the y 's will match up

If we add, we get

If we subtract the second equation from the third equation we get

and we get the same system of two equations in two unknowns we get in the substitiution method, so we will get the same answer as we got there. Let us continue with the addition method this time.

If we divide the bottom equation by 3

then we could double the new bottom equation to get the coefficients of z to match up.

If we now add, we get

so

Now that we know what x is we can solve for z by substituting this solution into any equation which contains only x and z. The simplest would be

This becomes

Transpose

or

Divide by the coefficient of the unknown,

Now that we have gotten the same answers for x and z as we did the last time, the same procedure will give us the same solution for y. Substitute these values for a and z into the second original equation and solve for y.

This now becomes.

which gives us the solution

which gives us the same solutions as the substitution method, Cramer's rule, row operations and inverse matrices, and which also checks

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