Math Problem Statement
Solve the system, if possible. negative −55xminus− 9yminus− z equals=negative −168 negative −88xminus− 15yminus− 2z equals=negative −266 66xplus+ 9yplus+ z equals=212Select the correct choice below and fill in any answer boxes within your choice. A. There is one solution. The solution is left parenthesis nothing comma nothing comma nothing right parenthesisenter your response here,enter your response here,enter your response here. (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expressions.) B. There are infinitely many solutions. The solutions are left parenthesis nothing comma nothing comma z right parenthesisenter your response here,enter your response here,z, where z is any real number. (Simplify your answers. Type expressions using z as the variable.) C. There is no solution.
Solution
We are given the system of three equations:
1) \ -55x - 9y - z &= -168 \\ 2) \ -88x - 15y - 2z &= -266 \\ 3) \ 66x + 9y + z &= 212 \end{aligned}$$ Let's solve this system step by step using elimination and substitution methods. ### Step 1: Eliminate $$z$$ From equations 1 and 3, we can eliminate $$z$$. Let's add the first and third equations: $$(-55x - 9y - z) + (66x + 9y + z) = -168 + 212$$ Simplifying: $$(-55x + 66x) + (-9y + 9y) + (-z + z) = 44$$ This simplifies to: $$11x = 44$$ Thus: $$x = 4$$ ### Step 2: Substitute $$x = 4$$ into the system Now substitute $$x = 4$$ into the first and second equations to solve for $$y$$ and $$z$$. #### Substituting into equation 1: $$-55(4) - 9y - z = -168$$ Simplifying: $$-220 - 9y - z = -168$$ $$-9y - z = 52 \quad \text{(Equation 4)}$$ #### Substituting into equation 2: $$-88(4) - 15y - 2z = -266$$ Simplifying: $$-352 - 15y - 2z = -266$$ $$-15y - 2z = 86 \quad \text{(Equation 5)}$$ ### Step 3: Solve for $$y$$ and $$z$$ We now have two equations with $$y$$ and $$z$$: $$-9y - z = 52 \quad \text{(Equation 4)}$$ $$-15y - 2z = 86 \quad \text{(Equation 5)}$$ Let's multiply equation 4 by 2 to eliminate $$z$$: $$2(-9y - z) = 2(52)$$ $$-18y - 2z = 104 \quad \text{(Equation 6)}$$ Now subtract equation 5 from equation 6: $$(-18y - 2z) - (-15y - 2z) = 104 - 86$$ Simplifying: $$-18y + 15y = 18$$ $$-3y = 18$$ Thus: $$y = -6$$ ### Step 4: Solve for $$z$$ Substitute $$y = -6$$ into equation 4: $$-9(-6) - z = 52$$ Simplifying: $$54 - z = 52$$ $$z = 2$$ ### Final Answer: We have found $$x = 4$$, $$y = -6$$, and $$z = 2$$. Thus, the solution to the system is: $$\boxed{(4, -6, 2)}$$ ### The correct choice is: **A. There is one solution. The solution is (4, -6, 2).** Would you like more details or have any questions? Here are 5 related questions for further practice: 1. Solve the system: $$2x + y + z = 7$$, $$3x - y - z = 2$$, and $$4x + 2y + z = 10$$. 2. Solve for $$x$$, $$y$$, and $$z$$ using substitution in the system: $$x - 2y + z = 3$$, $$2x + y + 3z = 8$$, $$-x + 4y + 2z = 1$$. 3. Solve the system of equations: $$x + 2y + z = 4$$, $$2x - y + 3z = 6$$, $$3x + y - z = 5$$. 4. Can a system of 3 linear equations have more than one solution? Explain why or why not. 5. What is the difference between solving a system of equations using substitution and elimination? **Tip**: When solving systems of equations, carefully choose a variable to eliminate first, and always check your work by substituting your solutions back into the original equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Systems of Equations
Elimination Method
Substitution Method
Formulas
Linear equation: ax + by + cz = d
Substitution: solving one equation for a variable and substituting in another
Elimination: combining equations to remove a variable
Theorems
Fundamental Theorem of Linear Systems (one solution, no solution, or infinitely many solutions)
Suitable Grade Level
Grades 10-12
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