Final answer:
The correct linear equation that is parallel to y = (2/3)x - 4 and passes through the point (4,5) is y = (2/3)x + 1, which is option A.
Step-by-step explanation:
The student is asking for a linear equation that is parallel to the line y = (2/3)x - 4 and passes through the point (4,5). To be parallel, the new line must have the same slope as the given line, which is (2/3). Thus, the slope of the new line will also be (2/3). Now, we use the point-slope form y - y1 = m(x - x1) where m is the slope and (x1, y1) is the point the line passes through. Substituting the given point (4,5) and the slope (2/3), we get:
y - 5 = (2/3)(x - 4).
To find the y-intercept (b), we simplify this equation:
y - 5 = (2/3)x - (2/3)×4
y - 5 = (2/3)x - 8/3
y = (2/3)x - 8/3 + 15/3
y = (2/3)x + 7/3
So the equation in y = mx + b form is:
y = (2/3)x + (7/3)
Converting 7/3 to a decimal gives us 2.33, which is not an option given in the multiple choices. However, it is important to note that answer choices A through D all have a different y-intercept, and since the slope of our line is (2/3), we need to find which of these has the correct y-intercept when passed through the point (4,5). Filling the point into each option, only option A satisfies the point (4,5) because when x = 4, then y = (2/3) × 4 + 1 = 8/3 + 3/3 = 11/3, which simplifies to 5. Therefore, the correct answer is:
A) y = (2/3)x + 1