To solve the given system of equations using the substitution method, we have the following equations:
- \( 3x - 3y + 4z = -23 \) (Equation 1)
- \( 2x + 4y - 6z = 50 \) (Equation 2)
- \( 4x - y + z = 25 \) (Equation 3)
Step 1: Solve one equation for one variable.
Let's solve Equation 3 for \( z \):
\[ z = 25 - 4x + y \tag{From Eq. 3} \]
Step 2: Substitute \( z \) in the other equations.
Now we'll substitute \( z \) in Equations 1 and 2.
Substituting into Equation 1:
\[ 3x - 3y + 4(25 - 4x + y) = -23 \]
Expanding this:
\[ 3x - 3y + 100 - 16x + 4y = -23 \]
Combining like terms:
\[ -13x + y + 100 = -23 \]
Solving for \( y \):
\[ y - 13x = -123 \]
Thus,
\[ y = 13x - 123 \tag{Equation 4} \]
Now substituting into Equation 2:
\[ 2x + 4(13x - 123) - 6(25 - 4x + (13x - 123)) = 50 \]
Expanding this:
\[ 2x + 52x - 492 - 6(25 - 4x + 13x - 123) = 50 \]
Calculating \( 6(25 - 4x + 13x - 123) \):
\[ 6(25 - 4x + 13x - 123) = 6(-98 + 9x) = -588 + 54x \]
Continuing to simplify:
\[ 2x + 52x - 492 + 588 - 54x = 50 \]
Combining like terms:
\[ 0x + 96 = 50 \]
This simplifies to \( 96 = 50 \), which is not true.
Conclusion:
Since we reached a contradiction, this means the system of equations has no solutions.
Thus, the answer is:
- This system has no solutions.
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