Math Problem Statement
Use the echelon method to solve the following system of two equations in two unknowns. Check your answer. 6 x plus 3 y equalsnegative 12 5 x minus 3 y equalsnegative 21
Solution
We are given the system of linear equations:
6x + 3y &= -12 \\ 5x - 3y &= -21 \end{aligned}$$ To solve using the echelon method, we will follow these steps: ### Step 1: Write the augmented matrix for the system The augmented matrix for the system is: $$\begin{pmatrix} 6 & 3 & | & -12 \\ 5 & -3 & | & -21 \end{pmatrix}$$ ### Step 2: Perform row operations to reach echelon form We want the matrix in the form where we can easily back-substitute. First, let's make the leading coefficient of the first row 1 by dividing the entire first row by 6. $$R_1 = \frac{1}{6} R_1$$ This gives us: $$\begin{pmatrix} 1 & \frac{1}{2} & | & -2 \\ 5 & -3 & | & -21 \end{pmatrix}$$ Next, we'll eliminate the $$5$$ in the second row by subtracting $$5 \times R_1$$ from $$R_2$$: $$R_2 = R_2 - 5R_1$$ This gives us: $$\begin{pmatrix} 1 & \frac{1}{2} & | & -2 \\ 0 & -\frac{11}{2} & | & -11 \end{pmatrix}$$ ### Step 3: Solve the system using back-substitution From the second row, we can solve for $$y$$: $$-\frac{11}{2} y = -11$$ Multiplying both sides by $$-2/11$$: $$y = 2$$ Now substitute $$y = 2$$ into the first row: $$1x + \frac{1}{2}(2) = -2$$ Simplifying: $$x + 1 = -2 \implies x = -3$$ ### Step 4: Check the solution Substitute $$x = -3$$ and $$y = 2$$ into the original equations: $$6(-3) + 3(2) = -18 + 6 = -12 \quad \text{(True)}$$ $$5(-3) - 3(2) = -15 - 6 = -21 \quad \text{(True)}$$ Thus, the solution is correct. ### Final Answer: $$x = -3, \quad y = 2$$ Do you have any questions or would you like further details? ### Related Questions: 1. What is the general procedure for solving a system of equations using the echelon method? 2. How do you determine if a system of equations has no solution using row echelon form? 3. Can the echelon method be used for non-square systems of equations? 4. How does back-substitution work in Gaussian elimination? 5. What are the differences between Gaussian elimination and Gauss-Jordan elimination? ### Tip: When using the echelon method, always aim to have leading 1's in your rows, as this simplifies back-substitution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Echelon Form
Back-Substitution
Formulas
Augmented Matrix Representation
Row Operations: Row Scaling and Row Replacement
Echelon Form to solve linear systems
Theorems
Gaussian Elimination
Row Echelon Form
Back-Substitution
Suitable Grade Level
Grades 9-12
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