We can solve the system of equations using either substitution or elimination. Here are the given equations:

1. \( 5x - y = 19 \)
2. \( 13x - 3y = 51 \)

Let's solve it using **elimination**.

### Step 1: Multiply the first equation to align coefficients of \(y\)
We want the coefficients of \(y\) in both equations to be the same. So, multiply the first equation by 3:

\[
3(5x - y) = 3(19)
\]
\[
15x - 3y = 57
\]

Now, the system becomes:
1. \( 15x - 3y = 57 \)
2. \( 13x - 3y = 51 \)

### Step 2: Subtract the second equation from the first
By subtracting the second equation from the first, we eliminate \(y\):

\[
(15x - 3y) - (13x - 3y) = 57 - 51
\]
\[
2x = 6
\]

### Step 3: Solve for \(x\)
\[
x = \frac{6}{2} = 3
\]

### Step 4: Substitute \(x = 3\) into the first equation
Now substitute \(x = 3\) into the first equation to find \(y\):

\[
5(3) - y = 19
\]
\[
15 - y = 19
\]
\[
-y = 19 - 15
\]
\[
-y = 4 \quad \Rightarrow \quad y = -4
\]

### Final Solution:
The solution to the system is:
\[
x = 3, \quad y = -4
\]

Let me know if you'd like more details or have any questions!

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Here are 5 related questions:

1. What is the graphical interpretation of the solution \(x = 3, y = -4\)?
2. Can you solve this system of equations using substitution?
3. How would the solution change if the second equation was \(13x - 3y = 60\)?
4. What happens if the two equations describe parallel lines?
5. How do you solve a system of equations with three variables?

**Tip:** Elimination works best when you can easily align coefficients for one variable.

This page provides a step-by-step solution to the system of equations 5x - y = 19 and 13x - 3y = 51 using the elimination method. Learn how to align coefficients, eliminate variables, and substitute to find x = 3 and y = -4. Suitable for Grades 8-10 algebra students.

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