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You have uploaded a system of linear equations:

\[
\begin{cases}
3x - 5y = 4 \\
2x + 6y = 3
\end{cases}
\]

Let's solve this system of equations using the substitution or elimination method.

### Step 1: Multiply the equations to align coefficients
We will use the elimination method. Multiply the first equation by 2 and the second by 3, to make the coefficients of \(x\) the same:

\[
\text{First equation: } 2(3x - 5y = 4) \implies 6x - 10y = 8
\]
\[
\text{Second equation: } 3(2x + 6y = 3) \implies 6x + 18y = 9
\]

### Step 2: Subtract the equations
Now subtract the second equation from the first:

\[
(6x - 10y) - (6x + 18y) = 8 - 9
\]
This simplifies to:

\[
-28y = -1
\]
\[
y = \frac{-1}{-28} = \frac{1}{28}
\]

### Step 3: Solve for \(x\)
Substitute \(y = \frac{1}{28}\) into one of the original equations, say \(3x - 5y = 4\):

\[
3x - 5\left(\frac{1}{28}\right) = 4
\]
\[
3x - \frac{5}{28} = 4
\]
Multiply through by 28 to eliminate the fraction:

\[
28(3x) - 5 = 28(4)
\]
\[
84x - 5 = 112
\]
\[
84x = 117
\]
\[
x = \frac{117}{84} = \frac{39}{28}
\]

### Final solution:
\[
x = \frac{39}{28}, \quad y = \frac{1}{28}
\]

Would you like a more detailed explanation of any step?

Here are some follow-up questions to enhance understanding:
1. How do you know which method (substitution vs. elimination) is better for solving a system of equations?
2. Can you verify the solution by substituting \(x\) and \(y\) into both original equations?
3. How would the system change if one of the equations was non-linear?
4. Could this system have been solved graphically? What would the solution represent graphically?
5. How would you handle a system of three equations and three unknowns?

**Tip:** When solving systems of equations, elimination is often preferred if the coefficients of one variable can easily be made the same.

Learn how to solve the system of linear equations 3x - 5y = 4 and 2x + 6y = 3 using the elimination method. This step-by-step solution demonstrates multiplying and subtracting equations to find the values of x and y. Suitable for students in Grades 8-10.

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