Sure, let's walk through the detailed step-by-step solution to identify the equation that Berta could have found in step 1.
Given system of equations:
[tex]\[
\begin{cases}
4x + y = -5 \\
x - y = -5
\end{cases}
\][/tex]
### Step 1: Solve for y from the second equation
From the second equation [tex]\(x - y = -5\)[/tex], we can solve for [tex]\(y\)[/tex]:
[tex]\[y = x + 5\][/tex]
### Step 2: Substitute [tex]\(y\)[/tex] into the first equation
Now we substitute [tex]\(y = x + 5\)[/tex] into the first equation [tex]\(4x + y = -5\)[/tex]:
[tex]\[4x + (x + 5) = -5\][/tex]
### Step 3: Simplify the substituted equation
Combine like terms:
[tex]\[4x + x + 5 = -5\][/tex]
[tex]\[5x + 5 = -5\][/tex]
### Step 4: Isolate the variable [tex]\(x\)[/tex]
Subtract 5 from both sides:
[tex]\[5x = -10\][/tex]
This mean the potential equation Berta found in step 1 could have been:
[tex]\[5x = -10\][/tex]
Option: [tex]\[5x = -10\][/tex]
### Verification:
- From [tex]\(5x = -10\)[/tex], solving for [tex]\(x\)[/tex] we get [tex]\(x = -2\)[/tex].
- Substitute [tex]\(x = -2\)[/tex] back into the equation [tex]\(x - y = -5\)[/tex]:
[tex]\[-2 - y = -5\][/tex]
Solving for [tex]\(y\)[/tex], we get:
[tex]\[- y = -3 \implies y = 3\][/tex]
Berta's solution [tex]\((-2, 3)\)[/tex] checks out, confirming that the potential equation she found in step 1 is:
[tex]\[5x = -10\][/tex]
Thus, the correct option is:
[tex]\[\boxed{5x = -10}\][/tex]
