To determine which equation, in standard form, is equivalent to the given equation:
[tex]\[
(y + 7)^2 = -14(x - 1)
\][/tex]
we follow these steps:
### Step 1: Expand the given equation
Start by expanding the left side of the equation [tex]\((y + 7)^2\)[/tex]:
[tex]\[
(y + 7)^2 = y^2 + 14y + 49
\][/tex]
So, the equation now reads:
[tex]\[
y^2 + 14y + 49 = -14(x - 1)
\][/tex]
### Step 2: Distribute the right side
Next, distribute [tex]\(-14\)[/tex] on the right side of the equation:
[tex]\[
y^2 + 14y + 49 = -14x + 14
\][/tex]
### Step 3: Rearrange the equation to solve for [tex]\(x\)[/tex]
Add 14x to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
y^2 + 14y + 49 + 14x = 14
\][/tex]
Then, move constants to the right side:
[tex]\[
14x = -y^2 - 14y + 14 - 49
\][/tex]
Combine like terms:
[tex]\[
14x = -y^2 - 14y - 35
\][/tex]
Finally, divide each term by 14 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = -\frac{1}{14} y^2 - y - \frac{35}{14}
\][/tex]
Simplify the constant:
[tex]\[
x = -\frac{1}{14} y^2 - y - \frac{5}{2}
\][/tex]
### Step 4: Check against the given standard form equations
We compare the derived equation with the given options:
1. [tex]\(x = \frac{1}{14} y^2 - y + 5\)[/tex]
2. [tex]\(x = \frac{1}{14} y^2 + y + 2\)[/tex]
3. [tex]\(x = -\frac{1}{14} y^2 - y + \frac{5}{2}\)[/tex]
4. [tex]\(x = -\frac{1}{14} y^2 - y - \frac{5}{2}\)[/tex]
Clearly, the derived equation matches the fourth given option:
[tex]\[
x = -\frac{1}{14} y^2 - y - \frac{5}{2}
\][/tex]
### Conclusion
Thus, the equation in standard form that is equivalent to the given equation [tex]\((y + 7)^2 = -14(x - 1)\)[/tex] is:
[tex]\[
\boxed{-\frac{1}{14} y^2 - y - \frac{5}{2}}
\][/tex]
Therefore, the answer is:
[tex]\[
x = -\frac{1}{14} y^2 - y - \frac{5}{2}
\][/tex] which corresponds to option 4.
