Solve the system of equations.

3x−y=17

x+4y=10

To solve the system of equations:

\[
3x - y = 17
\]
\[
x + 4y = 10
\]

we can use either substitution or elimination. Here, let's solve it using substitution:

### Step 1: Express \( x \) in terms of \( y \) from the second equation.

\[
x + 4y = 10 \implies x = 10 - 4y
\]

### Step 2: Substitute \( x \) into the first equation.

\[
3(10 - 4y) - y = 17
\]

Expanding this:

\[
30 - 12y - y = 17
\]

Combine like terms:

\[
30 - 13y = 17
\]

### Step 3: Solve for \( y \).

\[
-13y = 17 - 30
\]
\[
-13y = -13
\]
\[
y = 1
\]

### Step 4: Find \( x \) using \( y = 1 \) in the expression for \( x \).

\[
x = 10 - 4(1) = 10 - 4 = 6
\]

### Final Solution:
\[
x = 6, \quad y = 1
\]

### Verification:
- **First equation:** \( 3(6) - 1 = 18 - 1 = 17 \) ✔
- **Second equation:** \( 6 + 4(1) = 6 + 4 = 10 \) ✔

### Solution:
\[
\boxed{x = 6, \quad y = 1}
\]

Do you want more details or have any questions?

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Here are 5 questions based on this solution:
1. What is the elimination method, and how would it be applied to this system?
2. How do we handle systems of equations if both variables cancel out completely?
3. What if the equations represented parallel lines—how would the solution change?
4. How can you use matrix methods like row reduction to solve this system?
5. What are the key differences between substitution and elimination methods?

**Tip:** Always verify your solution by substituting the values back into both original equations. This helps ensure that the solution is correct!

Learn how to solve the system of linear equations 3x − y = 17 and x + 4y = 10 using the substitution method. This step-by-step solution covers important algebraic concepts, including variable substitution and verification, ideal for students in Grades 8-10.

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