Math Problem Statement
7x-6y=-5, -14x+12y=10 solve with echelon method
Solution
To solve the system of equations using the echelon method, we aim to reduce the system to a form where we can easily back-substitute to find the solution. The system of equations is:
Step 1: Write the system in matrix form.
We represent the system as an augmented matrix:
7 & -6 & | & -5 \\ -14 & 12 & | & 10 \end{pmatrix}$$ ### Step 2: Use row operations to reduce the matrix to echelon form. First, we perform row operations to eliminate the $$x$$-term from the second row. **Operation: $$R_2 = R_2 + 2 \times R_1$$** This means we multiply the first row by 2 and add it to the second row: $$R_2 = (-14 + 2 \times 7) \quad (12 + 2 \times (-6)) \quad (10 + 2 \times (-5))$$ $$R_2 = (0, 0, 0)$$ The new matrix is: $$\begin{pmatrix} 7 & -6 & | & -5 \\ 0 & 0 & | & 0 \end{pmatrix}$$ ### Step 3: Interpret the echelon form. The second row indicates that it's an identity equation $$0 = 0$$, which means it's consistent but gives no new information. The first row still represents a valid equation: $$7x - 6y = -5$$ ### Step 4: Solve for $$x$$ and $$y$$. From the first row: $$7x - 6y = -5$$ We can solve this equation for one variable in terms of the other. Let's solve for $$x$$: $$x = \frac{-5 + 6y}{7}$$ This means the solution is parametric, where $$y$$ can take any value and $$x$$ depends on that value. Thus, the solution set is: $$x = \frac{-5 + 6y}{7}, \quad y = y \quad (\text{where } y \in \mathbb{R})$$ This represents an infinite number of solutions, as the two equations are essentially the same. Would you like more details or have any questions? ### Related Questions: 1. What are the advantages of using the echelon method over substitution? 2. How would this solution change if there was a unique solution? 3. Can echelon form help in solving larger systems of equations? 4. What does it mean if we get an inconsistent system in echelon form? 5. How does this parametric solution represent a line? **Tip:** In systems with dependent equations, there are often infinitely many solutions forming a line or plane in higher dimensions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Echelon Form
Parametric Solutions
Formulas
R2 = R2 + 2 * R1
x = (-5 + 6y) / 7
Theorems
Row Operations
Echelon Form
Dependent Equations
Suitable Grade Level
Grades 10-12
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