Final answer:
The student needs to determine the equation of a line parallel to another given line and passing through a certain point. By establishing that parallel lines share the same slope and using the given point to find the y-intercept, the student can formulate the equation of the desired line.
Step-by-step explanation:
The subject of this question is Mathematics, and it seems tailored for High School students, particularly those studying algebra and the concept of linear equations. The student is asking for the equation of a line that is parallel to a given line and passes through a specified point.
To find an equation of a line parallel to the given line 254x - 5y = 25, we first need to rewrite the given line in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. We get the slope from the given line and use the fact that parallel lines have identical slopes. We then use the given point (5,4) to solve for the y-intercept of our new line using the equation y = mx + b.
Steps to Derive the Equation
- Rewrite the given equation in slope-intercept form: 254x - 5y = 25 to y = (254/5)x - 5.
- Identify the slope of the given line: The slope (m) is 254/5.
- Since parallel lines have the same slope, the slope of the new line will also be 254/5.
- Use the slope of the parallel line and the point (5,4) to find the y-intercept (b) using the formula y = mx + b: 4 = (254/5)*(5) + b.
- Solve for b, which gives us the y-intercept of the new line.
- Write the final equation of the line that is parallel to the given line and passes through (5,4).
This process applies the concept of linearity and the characteristics of parallel lines within a Cartesian coordinate system. We see that parallel lines have equivalent slopes by examining equations such as Y2 = -173.5 + 4.83x − 2(16.4) and Y3 = -173.5 + 4.83x + 2(16.4), which, despite different y-intercepts, maintain the same slope as their line of best fit. This consistency in slopes is a crucial property used to solve the problem.