Question

Solve the system of equations:
X+y+z=5
-3x-4y+4z=15
2x-y-4z=-8

Answer 1

(x, y, z) = (3, -2, 4)

Explanation:

The attachment shows a solution using a graphing calculator where the first equation is solved for z, and that expression is substituted into each of the other two equations. The solution to that system is (x, y) = (3, -2). Using these values in the expression for z, we find ...

z = 5 -(3 +(-2)) = 4

The solution is (x, y, z) = (3, -2, 4).

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Your graphing calculator and/or online tools can give you the reduced row echelon form of the augmented matrix representing the system:

`\left[\begin{array}c1&1&1&5\\-3&-4&4&15\\2&-1&-4&-8\end{array}\right]`

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If you're solving by hand, you can do the substitution described above to eliminate z from one equation. You can eliminate z from another equation by adding the last two:

-3x -4y +4(5 -x -y) = 15 ⇒ -7x -8y = -5

(-3x -4y +4z) +(2x -y -4z) = (15) +(-8) ⇒ -x -5y = 7

This pair of equations can be solved for x and y in any of the usual ways. Using the second equation to substitute for x, for example, gives you ...

-7(-5y -7) -8y = -5 ⇒ 27y = -54 ⇒ y = -2

x = -5(-2) -7 = 3

z = 5 -(3) -(-2) = 4

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Additional comment

If all that is needed is a solution, we prefer a graphical or matrix approach. These take about the same amount of time. Both require a little additional effort to determine exact values when the solutions are not integers.