What is an equivalent equation solved for y2? y2 = mx2 – x1 y1 y2 = mx2 – x1 – y1 y2 = m(x2 – x1) y1 y2 = m(x2 – x1) – y1
The Correct Answer and Explanation is :
To determine which equation is equivalent to (y_2), let’s start with the given options and analyze them step-by-step:
- Starting Equation: The original equation is (y_2 = mx_2 – x_1y_1).
- Options:
- (y_2 = mx_2 – x_1 – y_1)
- (y_2 = m(x_2 – x_1))
- (y_2 = m(x_2 – x_1) – y_1)
Evaluating Each Option:
- First Option: (y_2 = mx_2 – x_1 – y_1)
This option is not equivalent to the original equation. The term (-y_1) is introduced without justification, making this equation distinct from the original. - Second Option: (y_2 = m(x_2 – x_1))
This option rewrites (y_2) in a form based on the slope-intercept concept but removes any influence of (y_1) and (x_1) inappropriately. Thus, it does not hold equivalent value to the original equation. - Third Option: (y_2 = m(x_2 – x_1) – y_1)
This equation can be rearranged to show its equivalence to the original. Start with (y_2 = m(x_2 – x_1) – y_1):
- Adding (y_1) to both sides gives:
[
y_2 + y_1 = m(x_2 – x_1)
] - This shows a relationship involving both (y_2) and (y_1), which aligns with the concept of a slope in the coordinate system where (m) is the slope, and the differences in (x) values are accounted for.
Conclusion:
After evaluating the options, the equivalent equation solved for (y_2) is:
[ y_2 = m(x_2 – x_1) – y_1 ]
This option correctly captures the relationship defined by the original equation while maintaining the role of (y_1). Understanding these transformations is crucial in algebra, particularly in linear equations where slope-intercept form often plays a significant role. By manipulating the equations, we can derive relationships and understand how different variables interact within a given context.