To find the value of monthly sales where both companies have the same income, we set the two equations equal to each other:
- \( y = 500x + 4000 \) (Company A)
- \( y = 250x + 5000 \) (Company B)
Setting them equal to find the point of intersection:
\[ 500x + 4000 = 250x + 5000 \]
Now, we solve for \( x \):
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Subtract \( 250x \) from both sides:
\[ 500x - 250x + 4000 = 5000 \]
which simplifies to:
\[ 250x + 4000 = 5000 \]
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Subtract \( 4000 \) from both sides:
\[ 250x = 1000 \]
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Finally, divide by \( 250 \):
\[ x = \frac{1000}{250} = 4 \]
Thus, the value of monthly sales \( x \) where both companies pay the same income is \( 4 \) (which could represent thousands, depending on the context).
To confirm, we can substitute \( x = 4 \) back into both equations to check if they yield the same \( y \):
For Company A:
\[ y = 500(4) + 4000 = 2000 + 4000 = 6000 \]
For Company B:
\[ y = 250(4) + 5000 = 1000 + 5000 = 6000 \]
Both companies will have an income of \( 6000 \) at \( x = 4 \). Thus, the monthly sales value at which both companies have the same income is \( 4 \) (thousands).