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


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

\[
\begin{aligned}
1. & \ 4x + y = 13 \\
2. & \ 5x - y = 14
\end{aligned}
\]

### 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.

Use the echelon method to solve the following system of two equations in two unknowns. Check your answer. 4x + y = 13, 5x - y = 14.

Learn how to solve the system of equations 4x + y = 13 and 5x - y = 14 using the echelon method. This step-by-step solution demonstrates matrix row reduction and substitution, ideal for students in Grades 9-11.

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