Solving Linear System using Echelon Method: 4x + 5y = -2, 5x

Math Problem Statement

Use the echelon method to solve the system of equations: 4x + 5y = -2 and 5x - 4y = 18.

The problem asks to solve the following system of linear equations using the echelon method:

5x - 4y = 18$$ ### Step-by-step Solution: 1. **Form the augmented matrix** for the system: $$\begin{pmatrix} 4 & 5 & | & -2 \\ 5 & -4 & | & 18 \end{pmatrix}$$ 2. **Perform row operations** to get it into echelon form. First, multiply the first row by $$5$$ and the second row by $$4$$ so that the leading coefficients are the same: $$\begin{pmatrix} 20 & 25 & | & -10 \\ 20 & -16 & | & 72 \end{pmatrix}$$ 3. **Subtract Row 1 from Row 2** to eliminate $$x$$ in the second row: $$\begin{pmatrix} 20 & 25 & | & -10 \\ 0 & -41 & | & 82 \end{pmatrix}$$ 4. Solve for $$y$$ in the second row: $$-41y = 82 \implies y = \frac{82}{-41} = -2$$ 5. **Substitute** $$y = -2$$ into the first equation to solve for $$x$$: $$4x + 5(-2) = -2 \implies 4x - 10 = -2 \implies 4x = 8 \implies x = 2$$ Thus, the solution is $$x = 2$$ and $$y = -2$$, so the ordered pair is: $$(2, -2)$$ ### Final Answer: Option A: The solution is $$(2, -2)$$. Would you like further clarification or details? Here are 5 questions related to this topic: 1. What is the echelon form of a matrix? 2. How can we use row operations to solve systems of equations? 3. How does the echelon method differ from Gaussian elimination? 4. Can every system of linear equations be solved using the echelon method? 5. What are some real-life applications of solving systems of linear equations? **Tip**: Always verify your final solution by substituting the values of $$x$$ and $$y$$ back into the original equations.

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Algebra
Linear Equations
Echelon Method
Matrix Operations

Row operations on augmented matrices
x = (value derived from the equations)
y = (value derived from the equations)

Echelon Form
Matrix Row Reduction

Grades 10-12

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