Algebra 2 Common Core

Published by Prentice Hall
ISBN 10: 0133186024
ISBN 13: 978-0-13318-602-4

Chapter 3 - Linear Systems - 3-5 Systems With Three Variables - Practice and Problem-Solving Exercises - Page 171: 15

Answer

$(1, -4, 3)$

Work Step by Step

Label the original equations first: 1. $3x + 3y + 6z = 9$ 2. $2x + y + 3z = 7$ 3. $x + 2y - z = -10$ Add equations $1$ and $2$ and modify them such that one variable is the same in both equations but differs only in sign. Multiply equation $2$ by $-3$: 4. $-3(2x + y + 3z) = -3(7)$ Use distributive property: 4. $-6x - 3y - 9z = -21$ Combine equations $1$ and $4$: 1. $3x + 3y + 6z = 9$ 4. $-6x - 3y - 9z = -21$ Add the equations: 5. $-3x - 3z = -12$ Combine equations $2$ and $3$ to eliminate the $y$ variable. Modify equation $2$ by multiplying it by $-2$: 6. $-2(2x + y + 3z) = -2(7)$ Use distributive property: 6. $-4x - 2y - 6z = -14$ Combine equations $3$ and $6$: 3. $x + 2y - z = -10$ 6. $-4x - 2y - 6z = -14$ Add the equations together: 7. $-3x - 7z = -24$ Combine equations $5$ and $7$, but first, one variable must be eliminated. Multiply equation $5$ by $-1$: 8. $-1(-3x - 3z) = -1(-12)$ Use distributive property: 8. $3x + 3z = 12$ Combine equations $7$ and $8$: 7. $-3x - 7z = -24$ 8. $3x + 3z = 12$ Add the two equations: $-4z = -12$ Divide each side of the equation by $-4$: $z = 3$ Substitute this $z$ value into equation $8$ to find the value for $x$: 8. $3x + 3(3) = 12$ Multiply to simplify: $3x + 9 = 12$ Subtract $9$ from both sides of the equation: $3x = 3$ Divide both sides by $3$: $x = 1$ Substitute the values for $x$ and $z$ into one of the original equations to find $y$. Let's use equation $3$: 3. $1 + 2y - 3 = -10$ Multiply to simplify: $1 + 2y - 3 = -10$ Combine like terms on the left side of the equation: $2y - 2 = -10$ Add $2$ to each side of the equation: $2y = -8$ Divide both sides by $2$: $y = -4$ The solution is $(1, -4, 3)$. To check the solution, plug in the three values into one of the original equations. Use equation $2$: $2(1) + (-4) + 3(3) = 7$ $2 - 4 + 9 = 7$ $7 = 7$ The sides are equal to one another; therefore, the solution is correct.
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