To determine if the equations \( y+6.5=3(x+1.5) \) and \( y−11.5=2(x+3.5) \) represent the same line, we need to convert both equations to slope-intercept form (\( y = mx + b \)) and then compare their slopes and y-intercepts.
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For the first equation: \[ y + 6.5 = 3(x + 1.5) \] Distributing the \( 3 \): \[ y + 6.5 = 3x + 4.5 \] Subtracting \( 6.5 \) from both sides: \[ y = 3x + 4.5 - 6.5 \] \[ y = 3x - 2 \]
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For the second equation: \[ y - 11.5 = 2(x + 3.5) \] Distributing the \( 2 \): \[ y - 11.5 = 2x + 7 \] Adding \( 11.5 \) to both sides: \[ y = 2x + 7 + 11.5 \] \[ y = 2x + 18.5 \]
Now we can compare the two slope-intercept forms we found:
- The first equation converts to \( y = 3x - 2 \).
- The second equation converts to \( y = 2x + 18.5 \).
Since the slopes are different (3 versus 2), the two equations represent different lines.
Thus, the correct answer is: No. They cannot possibly represent the same line because they have different slopes.