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Math  /  Algebra

Question−3x−2y+3z=5x=82x−2y−2z=20\begin{array}{l}-3 x-2 y+3 z=5 \\ x=8 \\ 2 x-2 y-2 z=20\end{array}

Studdy Solution

STEP 1

1. We have a system of three equations with three variables: x x , y y , and z z .
2. The system is: $ \begin{align*} -3x - 2y + 3z &= 5 \\ x &= 8 \\ 2x - 2y - 2z &= 20 \end{align*} \]
3. We will use substitution and simplification to solve the system.

STEP 2

1. Substitute the known value of x x into the other equations.
2. Simplify the resulting equations.
3. Solve for the remaining variables y y and z z .

STEP 3

Substitute x=8 x = 8 into the first equation:
−3(8)−2y+3z=5 -3(8) - 2y + 3z = 5
Substitute x=8 x = 8 into the third equation:
2(8)−2y−2z=20 2(8) - 2y - 2z = 20

STEP 4

Simplify the first equation:
−24−2y+3z=5 -24 - 2y + 3z = 5
Add 24 to both sides:
−2y+3z=29 -2y + 3z = 29
Simplify the third equation:
16−2y−2z=20 16 - 2y - 2z = 20
Subtract 16 from both sides:
−2y−2z=4 -2y - 2z = 4

STEP 5

Now solve the system of two equations with two variables:
1. −2y+3z=29-2y + 3z = 29
2. −2y−2z=4-2y - 2z = 4
Subtract the second equation from the first:
(−2y+3z)−(−2y−2z)=29−4 (-2y + 3z) - (-2y - 2z) = 29 - 4
This simplifies to:
5z=25 5z = 25
Divide both sides by 5:
z=5 z = 5
Substitute z=5 z = 5 back into the second equation:
−2y−2(5)=4 -2y - 2(5) = 4
Simplify:
−2y−10=4 -2y - 10 = 4
Add 10 to both sides:
−2y=14 -2y = 14
Divide by -2:
y=−7 y = -7
The solution to the system is:
x=8,y=−7,z=5 x = 8, \quad y = -7, \quad z = 5

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