Question

Which ordered pair is a solution to the system of equations?

Note: Use the substitution method.

{x=14y+2−4x+3y=4



Responses

(72, 6 )
begin ordered pair 7 over 2 comma 6 end ordered pair

(−112, −263 )
begin ordered pair negative fraction 11 over 2 end fraction comma negative fraction 26 over 3 end fraction end ordered pair

(−14, 1 )
begin ordered pair negative 1 forth comma 1 end ordered pair

(54, 3 )

Answer 1

To determine which ordered pair is a solution to the system of equations given as \( x = 14y + 2 \) and \( -4x + 3y = 4 \), we will use the substitution method.

1. **Substituting \( x \) into the second equation:**

From the first equation, we have:
\[
x = 14y + 2
\]
Now substitute \( x \) in the second equation:
\[
-4(14y + 2) + 3y = 4
\]
Distributing:
\[
-56y - 8 + 3y = 4
\]
Combine like terms:
\[
-53y - 8 = 4
\]
Add 8 to both sides:
\[
-53y = 12
\]
Divide by -53:
\[
y = -\frac{12}{53}
\]

2. **Finding \( x \):**
Substitute \( y \) back into the first equation to find \( x \):
\[
x = 14\left(-\frac{12}{53}\right) + 2
\]
\[
x = -\frac{168}{53} + 2
\]
Converting 2 to a fraction:
\[
x = -\frac{168}{53} + \frac{106}{53} = -\frac{62}{53}
\]

So, the solution in decimal form is:
\[
\left(x, y\right) = \left(-\frac{62}{53}, -\frac{12}{53}\right)
\]

Now we will check each of the ordered pairs provided in the responses:

- \( (72, 6) \)
- \( \left(-\frac{11}{2}, -\frac{26}{3}\right) \)
- \( (-14, 1) \)
- \( (54, 3) \)

### Testing the responses:

1. **Testing \( (72, 6) \):**
- For \( x = 72 \) and \( y = 6 \):
- First equation: \( 72 = 14(6) + 2 \rightarrow 72 = 84 + 2 \) (not true)

2. **Testing \( \left(-\frac{11}{2}, -\frac{26}{3}\right) \):**
- Substitute \( y = -\frac{26}{3} \):
- First equation: \( -\frac{11}{2} = 14\left(-\frac{26}{3}\right) + 2 \)
- Calculate:
\[
14\left(-\frac{26}{3}\right) = -\frac{364}{3}
\]
\[
-\frac{11}{2} = -\frac{364}{3} + 2
\]
Convert 2 to a fraction:
\[
-\frac{364}{3} + \frac{6}{3} = -\frac{358}{3} \quad \text{(not true)}
\]

3. **Testing \( (-14, 1) \):**
- Substitute \( y = 1 \):
- First equation:
\[
-14 = 14(1) + 2 - (-14)
\]
- Not true.

4. **Testing \( (54, 3) \):**
- Substitute \( y = 3 \):
- First equation:
\[
54 = 14(3) + 2 \rightarrow 54 = 42 + 2 \quad \text{(not true)}
\]

### Conclusion:
None of the provided ordered pairs satisfy the system of equations. Please recheck the ordered pairs to see if they match the calculations given here.