Math Problem Statement

Use the echelon method to solve the following system of two equations in two unknowns. Check your answer: 2x + y = 8, 3x - y = 7.

Solution

The given system of equations is:

2x + y &= 8 \\ 3x - y &= 7 \end{aligned}$$ We are instructed to use the echelon method (Gaussian elimination) to solve this system. ### Step 1: Convert the system into an augmented matrix. $$\begin{bmatrix} 2 & 1 & | & 8 \\ 3 & -1 & | & 7 \end{bmatrix}$$ ### Step 2: Perform row operations to create an upper triangular matrix (echelon form). **Row 1 (R1):** $$\begin{bmatrix} 2 & 1 & | & 8 \end{bmatrix}$$ **Row 2 (R2):** $$\begin{bmatrix} 3 & -1 & | & 7 \end{bmatrix}$$ We will first eliminate the $$x$$-term in Row 2. To do this, we multiply Row 1 by $$\frac{3}{2}$$ and subtract it from Row 2: $$R2 \to R2 - \frac{3}{2} \times R1$$ Performing the operation: $$\begin{aligned} R1 &= \begin{bmatrix} 2 & 1 & | & 8 \end{bmatrix} \\ R2 &= \begin{bmatrix} 3 & -1 & | & 7 \end{bmatrix} - \frac{3}{2} \times \begin{bmatrix} 2 & 1 & | & 8 \end{bmatrix} \\ R2 &= \begin{bmatrix} 3 & -1 & | & 7 \end{bmatrix} - \begin{bmatrix} 3 & 1.5 & | & 12 \end{bmatrix} \\ R2 &= \begin{bmatrix} 0 & -2.5 & | & -5 \end{bmatrix} \end{aligned}$$ ### Step 3: Solve the system from the echelon form. The matrix is now: $$\begin{bmatrix} 2 & 1 & | & 8 \\ 0 & -2.5 & | & -5 \end{bmatrix}$$ From Row 2: $$-2.5y = -5 \quad \Rightarrow \quad y = 2$$ Now substitute $$y = 2$$ into Row 1: $$2x + y = 8 \quad \Rightarrow \quad 2x + 2 = 8 \quad \Rightarrow \quad 2x = 6 \quad \Rightarrow \quad x = 3$$ ### Step 4: Conclusion The solution is $$x = 3$$ and $$y = 2$$, so the ordered pair is: $$\boxed{(3, 2)}$$ Would you like any further details on this process? ### 5 Related Questions: 1. How do you perform row reduction to echelon form in a larger system of equations? 2. What is the significance of the echelon form in solving systems of equations? 3. Can Gaussian elimination be used for systems with more than two equations and unknowns? 4. What happens if there are infinitely many solutions in a system of equations? 5. How can systems of equations be solved using matrix inverses instead of Gaussian elimination? ### Tip: Always verify your solution by substituting it back into the original system of equations to check for consistency.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Systems
Gaussian Elimination

Formulas

Row operations to achieve echelon form
Solving linear equations by substitution after row reduction

Theorems

Gaussian elimination method
Echelon form in linear algebra

Suitable Grade Level

Grades 8-10

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