Solve the system of equations: 3x + 4y + 2z = 8, 2x - 3y - z = -3, x + 2y + z = 4.
Understand the Problem
The question is asking to solve a system of linear equations. Specifically, it presents three equations involving variables x, y, and z that need to be solved simultaneously.
Answer
The values for the variables are $x = \frac{3}{8}$, $y = \frac{1}{8}$, and $z = \frac{27}{8}$.
Answer for screen readers
The solution is: $$ x = \frac{3}{8}, \quad y = \frac{1}{8}, \quad z = \frac{27}{8}. $$
Steps to Solve
- List the Given Equations
The equations provided are: $$
- \quad 3x + y + 2z = 8 $$ $$
- \quad 2x - 3y - z = -3 $$ $$
- \quad x + 2y + z = 4 $$
- Isolate a Variable
Let's isolate $z$ from equation (1): $$ 2z = 8 - 3x - y \implies z = \frac{8 - 3x - y}{2} $$
- Substitute and Solve
Now, substitute $z$ into equations (2) and (3).
For equation (2): $$ 2x - 3y - \left(\frac{8 - 3x - y}{2}\right) = -3 $$ Multiply through by 2 to eliminate the fraction: $$ 4x - 6y - (8 - 3x - y) = -6 $$ Simplify: $$ 4x - 6y - 8 + 3x + y = -6 $$ Combine like terms: $$ (4x + 3x) + (-6y + y) - 8 = -6 \ 7x - 5y = 2 \quad \text{(4)} $$
For equation (3): $$ x + 2y + \left(\frac{8 - 3x - y}{2}\right) = 4 $$ Multiply through by 2: $$ 2x + 4y + (8 - 3x - y) = 8 $$ Simplify: $$ 2x + 4y + 8 - 3x - y = 8 $$ Combine like terms: $$ (2x - 3x) + (4y - y) = 8 - 8 \ -x + 3y = 0 \quad \text{(5)} $$
- Solve the New System of Equations
Now, solve equations (4) and (5): $$ 7x - 5y = 2 \quad \text{(4)} \ -x + 3y = 0 \quad \text{(5)} $$ From equation (5), isolate $x$: $$ x = 3y \ $$ Substitute $x = 3y$ into equation (4): $$ 7(3y) - 5y = 2 \ 21y - 5y = 2 \ 16y = 2 \ y = \frac{1}{8} $$ Substituting back to find $x$: $$ x = 3\left(\frac{1}{8}\right) = \frac{3}{8} $$
- Substituting to Find z
Substituting $x$ and $y$ back into the expression for $z$: $$ z = \frac{8 - 3\left(\frac{3}{8}\right) - \frac{1}{8}}{2} = \frac{8 - \frac{9}{8} - \frac{1}{8}}{2} = \frac{8 - \frac{10}{8}}{2} = \frac{\frac{64 - 10}{8}}{2} = \frac{54/8}{2} = \frac{54}{16} = \frac{27}{8}. $$
The solution is: $$ x = \frac{3}{8}, \quad y = \frac{1}{8}, \quad z = \frac{27}{8}. $$
More Information
This method of solving systems of equations is known as substitution and can be used for linear systems with any number of variables. The values obtained can often be checked by substituting them back into the original equations to verify correctness.
Tips
- Forgetting to multiply through by the least common denominator when eliminating fractions.
- Miscalculating when combining like terms.
- Failing to substitute the value of one variable into all relevant equations after isolating it.
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