Math Problem Statement
Solve the system of equations:
1. x + y + z = -1
2. 3x + 5y + 4z = 2
3. 3x + 6y + 5z = 0
Solution
The system of linear equations from the image can be written as follows:
1. & \quad x + y + z = -1 \\ 2. & \quad 3x + 5y + 4z = 2 \\ 3. & \quad 3x + 6y + 5z = 0 \end{align*}$$ I will now solve this system using methods like substitution or elimination. Let's proceed. ### Step 1: Solve the first equation for one variable From the first equation, solve for $$x$$: $$x = -1 - y - z$$ ### Step 2: Substitute into the other two equations Substitute $$x = -1 - y - z$$ into the second and third equations. #### Substitute into the second equation: $$3(-1 - y - z) + 5y + 4z = 2$$ Simplifying: $$-3 - 3y - 3z + 5y + 4z = 2$$ $$-3 + 2y + z = 2$$ $$2y + z = 5 \quad \text{(Equation 4)}$$ #### Substitute into the third equation: $$3(-1 - y - z) + 6y + 5z = 0$$ Simplifying: $$-3 - 3y - 3z + 6y + 5z = 0$$ $$-3 + 3y + 2z = 0$$ $$3y + 2z = 3 \quad \text{(Equation 5)}$$ ### Step 3: Solve the new system Now we have two equations with two unknowns: 1. $$2y + z = 5$$ (Equation 4) 2. $$3y + 2z = 3$$ (Equation 5) Let's solve this system. #### Multiply the first equation by 2 to eliminate $$z$$: $$4y + 2z = 10$$ Subtract Equation 5 from this: $$(4y + 2z) - (3y + 2z) = 10 - 3$$ $$y = 7$$ ### Step 4: Substitute $$y = 7$$ back into one of the equations Substitute into Equation 4: $$2(7) + z = 5$$ $$14 + z = 5$$ $$z = -9$$ ### Step 5: Solve for $$x$$ Substitute $$y = 7$$ and $$z = -9$$ into the expression for $$x$$: $$x = -1 - y - z = -1 - 7 - (-9) = -1 - 7 + 9 = 1$$ ### Final Solution: $$x = 1, \quad y = 7, \quad z = -9$$ Would you like more details, or do you have any questions? Here are some related questions to extend your understanding: 1. How do you use substitution to solve systems of linear equations? 2. What is the elimination method for solving systems of equations? 3. Can systems of equations always be solved by substitution? 4. What are some real-life applications of solving systems of equations? 5. How does the determinant of a system relate to its solvability? **Tip:** Always check your solution by substituting the values of $$x$$, $$y$$, and $$z$$ back into the original system of equations to verify.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: Solve one equation for one variable, substitute into others
Elimination: Multiply equations to eliminate a variable
Theorems
Substitution and Elimination are methods used to solve systems of linear equations
Suitable Grade Level
Grades 9-12