Math Problem Statement
If [x y z] is a solution of the system of equations 7x+2y+z=1 3y−z=5 −3x+4y−2z=8, then the value of x+y+z is
Solution
We are given the system of linear equations:
7x + 2y + z &= 1 \quad \text{(Equation 1)} \\ 3y - z &= 5 \quad \text{(Equation 2)} \\ -3x + 4y - 2z &= 8 \quad \text{(Equation 3)} \end{aligned}$$ Our goal is to find the value of $$x + y + z$$. ### Step 1: Solve for $$z$$ using Equation 2 From Equation 2: $$3y - z = 5$$ Solve for $$z$$: $$z = 3y - 5$$ ### Step 2: Substitute $$z = 3y - 5$$ into the other two equations Now substitute $$z = 3y - 5$$ into Equation 1 and Equation 3. **Substitute into Equation 1:** $$7x + 2y + (3y - 5) = 1$$ Simplify: $$7x + 5y - 5 = 1$$ $$7x + 5y = 6 \quad \text{(Equation 4)}$$ **Substitute into Equation 3:** $$-3x + 4y - 2(3y - 5) = 8$$ Simplify: $$-3x + 4y - 6y + 10 = 8$$ $$-3x - 2y = -2$$ $$3x + 2y = 2 \quad \text{(Equation 5)}$$ ### Step 3: Solve the system of two equations We now solve the system of two equations: 1. $$7x + 5y = 6$$ 2. $$3x + 2y = 2$$ Multiply Equation 5 by 5 and Equation 4 by 2 to eliminate $$y$$: $$5(3x + 2y) = 5(2) \quad \Rightarrow \quad 15x + 10y = 10$$ $$2(7x + 5y) = 2(6) \quad \Rightarrow \quad 14x + 10y = 12$$ Now subtract the second equation from the first: $$(15x + 10y) - (14x + 10y) = 10 - 12$$ $$x = -2$$ ### Step 4: Substitute $$x = -2$$ back into one of the equations Substitute $$x = -2$$ into Equation 5: $$3(-2) + 2y = 2$$ $$-6 + 2y = 2$$ $$2y = 8$$ $$y = 4$$ ### Step 5: Find $$z$$ Substitute $$y = 4$$ into $$z = 3y - 5$$: $$z = 3(4) - 5 = 12 - 5 = 7$$ ### Step 6: Calculate $$x + y + z$$ Now that we know $$x = -2$$, $$y = 4$$, and $$z = 7$$, we can calculate: $$x + y + z = -2 + 4 + 7 = 9$$ ### Final Answer: The value of $$x + y + z$$ is $$\boxed{9}$$. --- Would you like more details on any part of the solution? Here are some related questions: 1. How do you solve systems of linear equations using substitution? 2. What is the method of elimination for solving systems of equations? 3. How do you solve a system of equations with three variables? 4. Can the value of $$x + y + z$$ change if the system of equations has no solution? 5. What would happen if we used matrices to solve this system? **Tip:** When solving systems of equations, always look for substitutions that can simplify the equations, like solving for one variable in terms of others!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Systems of Equations
Substitution Method
Elimination Method
Formulas
Substitution Method: Solve one equation for a variable and substitute it into other equations.
Elimination Method: Multiply equations to align coefficients and eliminate variables.
Theorems
Consistency and Solvability of Linear Systems
Basic Properties of Linear Equations
Suitable Grade Level
Grades 9-12