To determine if the equations \( y+3=5(x+7.75) \) and \( y−40.75=5(x−1) \) represent the same line, we will convert both equations into slope-intercept form \( y = mx + b \).
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For the first equation: \( y + 3 = 5(x + 7.75) \)
Distribute on the right side: \[ y + 3 = 5x + 38.75 \] Now, isolate \( y \) by subtracting 3 from both sides: \[ y = 5x + 38.75 - 3 \] \[ y = 5x + 35.75 \]
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For the second equation: \( y - 40.75 = 5(x - 1) \)
Again, distribute on the right side: \[ y - 40.75 = 5x - 5 \] Now, isolate \( y \) by adding \( 40.75 \) to both sides: \[ y = 5x - 5 + 40.75 \] \[ y = 5x + 35.75 \]
After converting both equations, we see that:
- The first equation simplifies to \( y = 5x + 35.75 \)
- The second equation also simplifies to \( y = 5x + 35.75 \)
Since both equations result in the same slope and y-intercept, they represent the same line.
Answer: Yes, because they both convert to \( y = 5x + 35.75 \) in slope-intercept form.