Math  /  Algebra

QuestionSolve the system of equations by substitution. 3x+3yz=9x=72xy+2z=15\begin{array}{l} 3 x+3 y-z=-9 \\ x=-7 \\ -2 x-y+2 z=15 \end{array}

Studdy Solution

STEP 1

1. We are given a system of three equations with three variables: x x , y y , and z z .
2. The goal is to find the values of x x , y y , and z z that satisfy all three equations simultaneously.
3. We will use the substitution method, which involves substituting the value of one variable into the other equations.

STEP 2

1. Identify the equation that can be directly substituted into the others.
2. Substitute the known value into the other equations.
3. Solve the resulting system of equations.
4. Verify the solution by substituting all values back into the original equations.

STEP 3

We are given x=7 x = -7 directly from the second equation. This is the value we will substitute into the other equations.

STEP 4

Substitute x=7 x = -7 into the first equation:
3(7)+3yz=9 3(-7) + 3y - z = -9
Simplify:
21+3yz=9 -21 + 3y - z = -9

STEP 5

Add 21 to both sides to simplify further:
3yz=12 3y - z = 12
This is our new equation (Equation 4).

STEP 6

Substitute x=7 x = -7 into the third equation:
2(7)y+2z=15 -2(-7) - y + 2z = 15
Simplify:
14y+2z=15 14 - y + 2z = 15

STEP 7

Subtract 14 from both sides to simplify further:
y+2z=1 -y + 2z = 1
This is our new equation (Equation 5).

STEP 8

We now have a system of two equations with two variables:
1. 3yz=12 3y - z = 12
2. y+2z=1 -y + 2z = 1

Solve Equation 5 for y y :
y=12z -y = 1 - 2z
y=1+2z y = -1 + 2z

STEP 9

Substitute y=1+2z y = -1 + 2z into Equation 4:
3(1+2z)z=12 3(-1 + 2z) - z = 12
Simplify:
3+6zz=12 -3 + 6z - z = 12
5z3=12 5z - 3 = 12

STEP 10

Add 3 to both sides:
5z=15 5z = 15
Divide by 5:
z=3 z = 3

STEP 11

Substitute z=3 z = 3 back into y=1+2z y = -1 + 2z :
y=1+2(3) y = -1 + 2(3)
y=1+6 y = -1 + 6
y=5 y = 5

STEP 12

Verify the solution x=7 x = -7 , y=5 y = 5 , z=3 z = 3 by substituting back into the original equations:
1. 3(7)+3(5)3=9 3(-7) + 3(5) - 3 = -9 21+153=9 -21 + 15 - 3 = -9 9=9 -9 = -9 (True)
2. x=7 x = -7 (Given)
3. 2(7)5+2(3)=15 -2(-7) - 5 + 2(3) = 15 145+6=15 14 - 5 + 6 = 15 15=15 15 = 15 (True)
The solution satisfies all equations.
The solution is:
x=7,y=5,z=3 x = -7, \, y = 5, \, z = 3

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