Final answer:
None of the given options is the correct standard form for the line through the points (5,3) and (-1,5). The correct equation in standard form is x + 3y = 14, which does not match any of the provided choices. The correct option is b.
Step-by-step explanation:
To write an equation in standard form for the points (5,3) and (-1,5), first we need to find the slope of the line that passes through these points using the formula:
m = (y2 - y1) / (x2 - x1).
Let's apply the formula:
m = (5 - 3) / (-1 - 5) = 2 / -6 = -1/3.
Now that we have the slope, we can use point-slope form to write the equation:
y - y1 = m(x - x1).
Let's use point (5,3):
y - 3 = (-1/3)(x - 5).
Now we will distribute the slope on the right side and move all terms to one side to get the standard form:
3y - 9 = -x + 5,
which simplifies to:
x + 3y - 5 = 9.
And, to match with one of the offered options, let's multiply everything by -1:
-x - 3y + 5 = -9,
which is equivalent to:
-x - 3y = -14.
Adding x to both sides gives us 0 = x - 14 or x = 14. The only option that matches this form is option b) 2x - 3y = 11. However, since 2x - 3y = 11 simplifies to x - 1.5y = 5.5, this option does not exactly match the equation we derived. Therefore, none of the options provided is the correct standard form for the given points.
Instead, the equation in standard form is -2x - 6y = -28, which simplifies to 2x + 6y = 28 after multiplying by -1 again. To match the coefficients of the given options, we can divide everything by 2, resulting in x + 3y = 14. The correct option is b.