Math Problem Statement

Solve the system of equations for all three variables: -3x - 7y + 2z = -5 -3x + 8y - 10z = 7 3x + y + 4z = -7

Solution

The system of equations provided in the image is:

-3x - 7y + 2z &= -5 \\ -3x + 8y - 10z &= 7 \\ 3x + y + 4z &= -7 \\ \end{aligned}$$ Let's solve this system step by step using methods like elimination or substitution. ### Step 1: Eliminate one variable We can try to eliminate $$x$$ first. Add the first and third equations: $$(-3x - 7y + 2z) + (3x + y + 4z) = -5 + (-7)$$ This simplifies to: $$-6y + 6z = -12 \quad \text{or} \quad -y + z = -2 \quad \text{(Equation A)}$$ Next, subtract the first equation from the second to eliminate $$x$$: $$(-3x + 8y - 10z) - (-3x - 7y + 2z) = 7 - (-5)$$ Simplifying this: $$8y + 7y - 10z - 2z = 12$$ $$15y - 12z = 12 \quad \text{(Equation B)}$$ ### Step 2: Solve the two-variable system Now we have two equations: 1. $$-y + z = -2$$ (Equation A) 2. $$15y - 12z = 12$$ (Equation B) We can solve these equations for $$y$$ and $$z$$. First, solve Equation A for $$z$$: $$z = y - 2$$ Substitute this into Equation B: $$15y - 12(y - 2) = 12$$ Simplifying: $$15y - 12y + 24 = 12$$ $$3y + 24 = 12$$ $$3y = -12$$ $$y = -4$$ ### Step 3: Find $$z$$ Substitute $$y = -4$$ into $$z = y - 2$$: $$z = -4 - 2 = -6$$ ### Step 4: Find $$x$$ Now substitute $$y = -4$$ and $$z = -6$$ into the first equation: $$-3x - 7(-4) + 2(-6) = -5$$ $$-3x + 28 - 12 = -5$$ $$-3x + 16 = -5$$ $$-3x = -21$$ $$x = 7$$ ### Solution: The solution to the system is: $$x = 7, \quad y = -4, \quad z = -6$$ Would you like a deeper explanation or more details about this solution? Here are five related questions to explore: 1. How would the solution change if one equation were altered slightly? 2. What other methods can be used to solve a system of three variables? 3. How can matrix methods, like Gaussian elimination, be applied here? 4. What are the geometric interpretations of the solution to a system of three equations? 5. How do you check if a solution is correct for a system of linear equations? **Tip**: Always verify your solution by substituting the values of $$x$$, $$y$$, and $$z$$ back into the original equations to ensure they satisfy all equations.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Elimination Method
Substitution Method

Formulas

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Theorems

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Suitable Grade Level

High School (Grades 10-12)

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