Answer
$(\frac{1}{2}, -3, 1)$
Work Step by Step
Label the original equations first:
1. $6q - r + 2s = 8$
2. $2q + 3r - s = -9$
3. $4q + 2r + 5s = 1$
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 $1$ by $3$:
4. $3(6q - r + 2s) = 3(8)$
Use distributive property:
4. $18q - 3r + 6s = 24$
Combine equations $2$ and $3$:
2. $2q + 3r - s = -9$
4. $18q - 3r + 6s = 24$
Add the equations:
5. $20q + 5s = 15$
Combine equations $1$ and $3$ to eliminate the $r$ variable. Modify equation $1$ by multiplying it by $2$:
6. $2(6q - r + 2s) = 2(8)$
Use distributive property:
6. $12q - 2r + 4s = 16$
Combine equations $3$ and $6$:
3. $4q + 2r + 5s = 1$
6. $12q - 2r + 4s = 16$
Add the equations together:
7. $16q + 9s = 17$
Combine equations $5$ and $7$, but first, one variable must be eliminated. Multiply equation $5$ by $9$ and equation $7$ by $-5$:
5. $9(20q + 5s) = 9(15)$
7. $-5(16q + 9s) = -5(17)$
Use distributive property:
5. $180q + 45s = 135$
7. $-80q + -45s = -85$
Add the two equations:
$100q = 50$
Divide each side of the equation by $100$:
$q = \frac{50}{100}$
Simplify the fraction:
$q = \frac{1}{2}$
Substitute this $q$ value into equation $5$ to find the value for $s$:
5. $20(\frac{1}{2}) + 5s = 15$
Multiply to simplify:
$10 + 5s = 15$
Subtract $10$ from both sides of the equation:
$5s = 5$
Divide both sides by $5$:
$s = 1$
Substitute the values for $q$ and $s$ into one of the original equations to find $a$. Let's use equation $2$:
2. $2(\frac{1}{2}) + 3r - 1 = -9$
Multiply to simplify:
$1 + 3r - 1 = -9$
Combine like terms on the left side of the equation:
$3r + 0 = -9$
Subtract $0$ from each side of the equation:
$3r = -9$
Divide both sides by $3$:
$r = -3$
The solution is $(\frac{1}{2}, -3, 1)$.
To check the solution, plug in the three values into one of the original equations. Use equation $2$:
$2(\frac{1}{2}) + 3(-3) - 1 = -9$
$1 -9 - 1 = -9$
$-9 = -9$
The sides are equal to one another; therefore, the solution is correct.
