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Berta worked to solve a system of equations.
[tex]\[
\begin{array}{c}
4x + y = -5 \\
x - y = -5
\end{array}
\][/tex]
Step 1: ??? Step 2: [tex]\( x = -2 \)[/tex] Step 3: [tex]\( 4(-2) + y = -5 \)[/tex] Step 4: [tex]\( -8 + y = -5 \)[/tex] Step 5: [tex]\( y = 3 \)[/tex]
Berta determined that the solution is [tex]\((-2, 3)\)[/tex].
Which could have been the equation Berta found in step 1?
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]
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