Question

Solve the system

`2x + y = 9 \\ x - y + 5z = 8 \\ 3y - 2z = 1`
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Answer 1

(x, y, z) = (4, 1, 1)

Explanation:

Your graphing calculator can do this easily. the augmented matrix for the system is ...

`\left[\begin{array}c2&1&0&9\\1&-1&5&8\\0&3&-2&1\end{array}\right]`

Row reduction tells you the solution is ...

(x, y, z) = (4, 1, 1)

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You can use the second equation to write an expression for x that will substitute into the first equation:

x = 8 +y -5z

2(8 +y -5z) +y = 9

3y -10z = -7 . . . . . . . subtract 16, simplify

Subtracting this from the third equation gives ...

(3y -2z) -(3y -10z) = (1) -(-7)

8z = 8

z = 1 . . . . . divide by 8

Substituting into the third equation, you have ...

3y -2(1) = 1

3y = 3 . . . . . . add 2

y = 1 . . . . . . . divide by 3

Substituting into the expression for x gives ...

x = 8 + 1 -5(1) = 4

The solution is (x, y, z) = (4, 1, 1).

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Additional comment

The "by hand" solution generally takes advantage of any opportunities that present themselves for elimination or substitution. It is ad hoc in that respect. You can use algorithmic methods, but they generally will involve more steps.

Once you have reduced the equations to a set of 2 in 2 variables, you could do their solution graphically, or by any of the usual means: substitution, elimination, Cramer's rule, other matrix methods.

Some graphing calculators can do the substitution for you. The attachment shows an example of this. This particular calculator supports equations only in x and y, so we have used y1 for the original y, and y for the original z. The expression for y1 comes from the first original equation. The solution point (4, 1) means x=4, z=1, and the value of y can be found from 9-2(4) = 1.