Which of these is equivalent to the equation below?
y=mx+b
A. b=y−mx
B. x=y−b/m
C. b=mx/y
D. x=y/mb
The Correct Answer and Explanation is:
To determine which option is equivalent to the equation (y = mx + b), we need to manipulate the original equation and analyze each option.
The equation (y = mx + b) represents a linear equation in slope-intercept form, where (m) is the slope of the line, (b) is the y-intercept, and (x) and (y) are the variables. To find an equivalent expression, we can solve for (b), (x), or (y).
Rearranging the equation:
- Isolate (b):
[
b = y – mx
]
This directly corresponds to option A: (b = y – mx).
Now, let’s examine the other options to see if they represent equivalent forms or not.
- Option B:
[
x = \frac{y – b}{m}
]
Rearranging (y = mx + b) gives us (y – b = mx), leading to (x = \frac{y – b}{m}). While this rearrangement is valid, it is not the same as the original equation; instead, it offers a way to find (x). - Option C:
[
b = \frac{mx}{y}
]
This is incorrect because it doesn’t follow from the original equation. Instead, it falsely suggests a direct relationship between (b), (m), (x), and (y). - Option D:
[
x = \frac{y}{mb}
]
This option is also incorrect and not derived from the original equation. It misrepresents the relationship between (x), (y), (m), and (b).
Conclusion:
The only correct option that is directly derived from the equation (y = mx + b) is A. (b = y – mx). This option correctly isolates (b), confirming its equivalence to the original equation while maintaining the relationships of the variables involved. Thus, option A is the only valid choice here.