# How to solve / set up three equations three unknowns with a matrix?

How do I solve these equations with a matrix to get $x$, $y$, and $z$? I'm unsure of how to start.

$$0.09x + 0.10y + 0.12z = 52,000 \\ x + y + z = 500,000 \\ y = 2.5x$$

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$$a x + b y + cz = d \\ e x + f y + g z = h \\ i x + j y + k z = \ell$$ becomes $$\pmatrix{a & b & c \\ e & f & g \\ i & j & k} \pmatrix{x\\y\\z} = \pmatrix{d \\ h \\ \ell}.$$ –  user2468 Sep 11 '12 at 22:31

First, get rid of the decimals by multiplying the first and third equations by 100.

Next, write the three rows of the matrix A as:

A = {{9, 10, 12}, {1, 1, 1}, {-250, 100, 0}}

Next, write the column matrix for the solution:

b = {5200000, 500000, 0}

Solve by multiplying the inverse of A.b

A^(-1).b = {100000, 250000, 150000}

Verify the answer for each equation and it checks.

HTH

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I got the same answer! Thank you! –  Grace C Sep 11 '12 at 22:54
Computing $A^{-1}$ is generally considered not to be as good as forming an augmented matrix and doing row reduction. –  Gerry Myerson Sep 12 '12 at 2:00