Solve the System of Equations 10x - 2y + 3z = 23; 2x + 10y - 5z = -33; 3x

Math Problem Statement

Solve the system of equations by Gauss Elimination method 10x-2y+3z=23; 2x+10y-5z=-33; 3x-4y+10z=41. without decimal values

Solution

To solve the system of equations using the Gauss Elimination method, we follow these steps:

Given system of equations:

10x - 2y + 3z &= 23 \quad \text{(Equation 1)} \\ 2x + 10y - 5z &= -33 \quad \text{(Equation 2)} \\ 3x - 4y + 10z &= 41 \quad \text{(Equation 3)} \end{aligned}$$ ### Step 1: Write the augmented matrix The augmented matrix for this system is: $$\begin{pmatrix} 10 & -2 & 3 & | & 23 \\ 2 & 10 & -5 & | & -33 \\ 3 & -4 & 10 & | & 41 \end{pmatrix}$$ ### Step 2: Perform row operations to get the matrix into row echelon form (REF) 1. **Eliminate the first column entries below the first pivot (10) using Row 1 (R1):** - $$R2 \rightarrow R2 - \frac{2}{10} \times R1 = R2 - \frac{1}{5} \times R1$$ - $$R3 \rightarrow R3 - \frac{3}{10} \times R1$$ After performing these operations, we get: $$\begin{pmatrix} 10 & -2 & 3 & | & 23 \\ 0 & \frac{52}{5} & -\frac{19}{5} & | & -\frac{244}{5} \\ 0 & -\frac{26}{5} & \frac{91}{10} & | & \frac{187}{10} \end{pmatrix}$$ 2. **Multiply the second row by 5/52 to simplify:** $$R2 \rightarrow \frac{5}{52} \times R2$$ The new second row is: $$\begin{pmatrix} 0 & 1 & -\frac{19}{52} & | & -\frac{61}{13} \end{pmatrix}$$ So the updated augmented matrix is: $$\begin{pmatrix} 10 & -2 & 3 & | & 23 \\ 0 & 1 & -\frac{19}{52} & | & -\frac{61}{13} \\ 0 & -\frac{26}{5} & \frac{91}{10} & | & \frac{187}{10} \end{pmatrix}$$ 3. **Eliminate the second column entry below the pivot in the second row:** - $$R3 \rightarrow R3 + \frac{26}{5} \times R2$$ After performing the operation: $$\begin{pmatrix} 10 & -2 & 3 & | & 23 \\ 0 & 1 & -\frac{19}{52} & | & -\frac{61}{13} \\ 0 & 0 & \frac{4037}{260} & | & \frac{7861}{260} \end{pmatrix}$$ ### Step 3: Back-substitution to find the values of the variables From the third row, solve for $$z$$: $$z = \frac{7861}{4037} = 1.947 \approx 2 \quad \text{(rounded without decimals)}$$ Now, substitute $$z$$ into the second equation: $$y - \frac{19}{52}(2) = -\frac{61}{13}$$ $$y = -\frac{61}{13} + \frac{38}{52} = -5 \quad \text{(rounded without decimals)}$$ Finally, substitute $$y$$ and $$z$$ into the first equation: $$10x - 2(-5) + 3(2) = 23$$ $$10x + 10 + 6 = 23 \rightarrow 10x = 7 \rightarrow x = \frac{7}{10} \approx 0.7 \quad \text{(rounded without decimals)}$$ ### Final Solution - $$x = 1$$ - $$y = -5$$ - $$z = 2$$ If you want to avoid rounding and solve exactly, replace the approximations in the back substitution with exact values. --- Would you like the detailed steps on how to refine the solution further without approximation? Here are five related questions: 1. How would the solution change if decimals were allowed? 2. What are the advantages of Gauss Elimination over other methods? 3. How can the Gauss-Jordan method be used to solve the same system? 4. What role does matrix rank play in solving systems of equations? 5. How can we determine if a system has no solution or infinite solutions? **Tip:** Gauss Elimination is particularly useful when you need a systematic approach to solving linear systems, especially when dealing with larger matrices.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Gauss Elimination Method
Systems of Equations

Formulas

Theorems

Suitable Grade Level

Advanced High School

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