Solve the system by the addition method.




3 x plus 14 y

3x+14y

=

9

9

4 x minus 14 y

4x−14y

=

12

12







**A.**The solution is enter your response here. ​(Type an ordered​ pair.)




**B.**There are infinitely many solutions.




**C.**There is no solution. 

We are asked to solve the system of equations using the addition method:

1. \( 3x + 14y = 9 \)
2. \( 4x - 14y = 12 \)

### Step 1: Add the two equations
To eliminate \( y \), we can add the two equations directly since the \( y \)-terms cancel each other out:

\[
(3x + 14y) + (4x - 14y) = 9 + 12
\]

Simplifying:

\[
3x + 4x = 9 + 12
\]

\[
7x = 21
\]

### Step 2: Solve for \( x \)

\[
x = \frac{21}{7} = 3
\]

### Step 3: Substitute \( x = 3 \) into one of the original equations

Substitute into the first equation \( 3x + 14y = 9 \):

\[
3(3) + 14y = 9
\]

\[
9 + 14y = 9
\]

Subtract 9 from both sides:

\[
14y = 0
\]

\[
y = \frac{0}{14} = 0
\]

### Final Solution:
The solution is the ordered pair \( (x, y) = (3, 0) \).

Thus, the answer is:

**A.** The solution is \( (3, 0) \).

---

Would you like a deeper explanation or details on any step?

Here are 5 questions for deeper understanding:

1. What happens if the coefficients of \( y \) do not cancel out?
2. How does the addition method compare with substitution in solving systems of equations?
3. Could we have solved the system by graphing the equations?
4. What are the criteria for systems of equations to have infinitely many solutions?
5. How would the solution change if we altered the constants on the right-hand side?

**Tip:** Always check your solution by substituting both values into the original equations to ensure they satisfy both!

Solve the system by the addition method: 3x + 14y = 9, 4x - 14y = 12.

Learn how to solve a system of linear equations using the addition method. The problem involves eliminating variables and solving for x and y. Detailed steps are provided for solving the system 3x + 14y = 9 and 4x - 14y = 12. Suitable for Grades 8-10.

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