You've provided two linear equations:

1. \(3x + 2y = 45\)
2. \(4x - y = 5\)

This is a system of linear equations in two variables, \(x\) and \(y\). We can solve this system using various methods, such as substitution, elimination, or matrix methods (like the determinant method). Let's use the **substitution method** here.

### Step 1: Solve the second equation for \(y\)
From the second equation:
\[
4x - y = 5
\]
\[
y = 4x - 5
\]

### Step 2: Substitute \(y\) in the first equation
Now, substitute \(y = 4x - 5\) into the first equation:
\[
3x + 2(4x - 5) = 45
\]
Simplify and solve for \(x\):
\[
3x + 8x - 10 = 45
\]
\[
11x - 10 = 45
\]
\[
11x = 55
\]
\[
x = 5
\]

### Step 3: Solve for \(y\)
Now that we know \(x = 5\), substitute it back into the expression for \(y\):
\[
y = 4(5) - 5
\]
\[
y = 20 - 5 = 15
\]

### Solution
The solution to the system is:
\[
x = 5, \quad y = 15
\]

This means the point \((5, 15)\) is the intersection of the two lines represented by the equations.

Would you like to explore another method to solve this system or have any other questions?

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Here are 5 related questions you might find interesting:

1. What are the advantages of using the substitution method over the elimination method?
2. How can you solve the system of equations using the elimination method?
3. What does it mean if the system of equations has no solution or infinitely many solutions?
4. How can you solve a system of linear equations using matrices?
5. What is the graphical interpretation of the solution to a system of linear equations?

**Tip:** Always check your solution by substituting the values of \(x\) and \(y\) back into the original equations to verify they satisfy both equations.

Learn how to solve the system of linear equations 3x + 2y = 45 and 4x - y = 5 using the substitution method. This problem covers key concepts in algebra, including systems of equations and the substitution method. Suitable for students in Grades 9-12.

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