To determine whether the equations represent the same line, we need to convert them into slope-intercept form (y = mx + b) and compare the resulting equations.
-
For the first equation: \( y - 7.5 = \frac{2}{1}(x + 3) \)
Start by distributing the 2: \[ y - 7.5 = 2(x + 3) \] \[ y - 7.5 = 2x + 6 \] Now, add 7.5 to both sides: \[ y = 2x + 6 + 7.5 \] \[ y = 2x + 13.5 \]
-
For the second equation: \( y + 20 = \frac{2}{1}(x - 1) \)
Again, distribute the 2: \[ y + 20 = 2(x - 1) \] \[ y + 20 = 2x - 2 \] Now, subtract 20 from both sides: \[ y = 2x - 2 - 20 \] \[ y = 2x - 22 \]
Now we have:
- The first equation in slope-intercept form is \( y = 2x + 13.5 \).
- The second equation in slope-intercept form is \( y = 2x - 22 \).
Since the slopes (coefficients of x) are the same but the y-intercepts (constants) are different, the equations represent different lines.
Therefore, the correct conclusion is:
No, because they both convert to different equations in slope-intercept form.