Math Problem Statement

Use the echelon method to solve the following system of two equations in two unknowns. Check your answer. 4 x plus y equals13 5 x minus y equals14

Solution

Let's solve the system of equations using the echelon method. The given system is:

1. & \ 4x + y = 13 \\ 2. & \ 5x - y = 14 \end{aligned}$$ ### Step 1: Write the system in augmented matrix form We represent the system as an augmented matrix: $$\begin{pmatrix} 4 & 1 & | & 13 \\ 5 & -1 & | & 14 \end{pmatrix}$$ ### Step 2: Eliminate the variable $$y$$ from the second row We want to create a zero in the second row, first column, by eliminating $$x$$ in the second equation. To do this, we'll perform row operations. - Multiply the first row by $$5$$ and the second row by $$4$$ to match the coefficients of $$x$$: $$R_1: \ 5(4x + y = 13) \implies 20x + 5y = 65$$ $$R_2: \ 4(5x - y = 14) \implies 20x - 4y = 56$$ Now subtract the second equation from the first: $$(20x + 5y) - (20x - 4y) = 65 - 56$$ $$9y = 9$$ $$y = 1$$ ### Step 3: Substitute $$y = 1$$ into one of the original equations Substitute $$y = 1$$ into the first equation $$4x + y = 13$$: $$4x + 1 = 13$$ $$4x = 12$$ $$x = 3$$ ### Step 4: Check the solution Substitute $$x = 3$$ and $$y = 1$$ into the second equation $$5x - y = 14$$: $$5(3) - 1 = 14$$ $$15 - 1 = 14$$ The solution is correct. ### Final Answer: $$x = 3, \quad y = 1$$ Would you like more details on any specific part of the solution? ### Related Questions: 1. How can we apply the echelon method for larger systems of equations? 2. What are the advantages of the echelon method over substitution or elimination? 3. How do we use matrix inversion to solve systems of equations? 4. Can we use the echelon method for non-linear systems of equations? 5. How does row reduction relate to finding the rank of a matrix? ### Tip: When using the echelon method, it's important to keep track of row operations carefully to avoid calculation errors.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Matrix Methods

Formulas

Echelon method
Row operations for matrices
Substitution in equations

Theorems

Linear independence of equations
Row reduction in matrices

Suitable Grade Level

Grades 9-11

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