Math Problem Statement

Solve the following system of linear equations: 1. x + 4y - 6z = -1 2. 2x - y + 2z = -7 3. -x + 2y - 4z = 5

Solution

The image contains a system of linear equations with three variables, xx, yy, and zz. The system is as follows:

1.\ & x + 4y - 6z = -1 \\ 2.\ & 2x - y + 2z = -7 \\ 3.\ & -x + 2y - 4z = 5 \end{aligned}$$ I will solve this system of equations using the method of substitution, elimination, or matrix operations. Let's proceed with solving the system. ### Step 1: Add Equation 1 and Equation 3 $$(x + 4y - 6z) + (-x + 2y - 4z) = -1 + 5$$ This simplifies to: $$6y - 10z = 4 \quad \text{(Equation 4)}$$ ### Step 2: Add Equation 2 and Equation 3 $$(2x - y + 2z) + (-x + 2y - 4z) = -7 + 5$$ This simplifies to: $$x + y - 2z = -2 \quad \text{(Equation 5)}$$ ### Step 3: Solve Equation 4 for y $$6y = 10z + 4$$ $$y = \frac{5z + 2}{3} \quad \text{(Equation 6)}$$ ### Step 4: Substitute Equation 6 into Equation 5 Substitute $$y = \frac{5z + 2}{3}$$ into $$x + y - 2z = -2$$: $$x + \frac{5z + 2}{3} - 2z = -2$$ Multiply the entire equation by 3 to eliminate the fraction: $$3x + 5z + 2 - 6z = -6$$ Simplify: $$3x - z = -8$$ $$z = 3x + 8 \quad \text{(Equation 7)}$$ ### Step 5: Substitute Equation 7 into Equation 6 Substitute $$z = 3x + 8$$ into $$y = \frac{5z + 2}{3}$$: $$y = \frac{5(3x + 8) + 2}{3}$$ Simplify: $$y = \frac{15x + 40 + 2}{3}$$ $$y = 5x + 14 \quad \text{(Equation 8)}$$ ### Step 6: Solve for x Now that we have $$z = 3x + 8$$ and $$y = 5x + 14$$, substitute these into any original equation to solve for $$x$$. Let's use Equation 1: $$x + 4(5x + 14) - 6(3x + 8) = -1$$ Expand and simplify: $$x + 20x + 56 - 18x - 48 = -1$$ $$3x + 8 = -1$$ $$x = -\frac{3}{4}$$ ### Step 7: Substitute x back to find y and z Now substitute $$x = -\frac{3}{4}$$ into the expressions for $$y$$ and $$z$$: $$y = 5\left(-\frac{3}{4}\right) + 14 = -\frac{15}{4} + \frac{56}{4} = \frac{41}{4}$$ $$z = 3\left(-\frac{3}{4}\right) + 8 = -\frac{9}{4} + \frac{32}{4} = \frac{23}{4}$$ ### Final Solution The solution to the system of equations is: $$\boxed{x = -\frac{3}{4},\ y = \frac{41}{4},\ z = \frac{23}{4}}$$ Would you like more details, or do you have any questions? Here are five related questions to expand on this topic: 1. How would you solve the system of equations using matrix operations? 2. Can you solve the system using the substitution method instead? 3. How can you determine if a system of equations is consistent or inconsistent? 4. What are the implications if a system of equations has infinitely many solutions? 5. How would you check the solution to ensure it is correct? **Tip:** When solving systems of equations, always check your solution by substituting the values back into the original equations to ensure consistency.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
System of Equations

Formulas

Substitution Method
Elimination Method

Theorems

Linear System Solution Theorem

Suitable Grade Level

Grades 10-12

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