(a) Find the equation of the line which passes through the points A(-2, 7) and B(2, -3)
(b) Given that \(\frac{5b - a}{8b + 3a} = \frac{1}{5}\) = find, correct to two decimal places, the value \(\frac{a}{b}\)
a) To find the equation of the line that passes through two points A and B:
The slope-point form of a line is given by:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is a point on the line. The slope of the line can be found using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values of x1, y1, x2, and y2, we get:
m = (-3 - 7) / (2 - (-2)) = -10 / 4 = -5/2
So, the equation of the line in slope-point form becomes:
y - 7 = -5/2 (x + 2)
Expanding the right-hand side, we get:
y - 7 = -5/2 x - 5
Adding 5 to both sides, we get:
y - 2 = -5/2 x
Dividing both sides by -5/2, we get:
2x + y = 9
This is the equation of the line that passes through the points A(-2, 7) and B(2, -3).
b) To solve for a/b:
We need to rearrange the equation:
\(\frac{5b - a}{8b + 3a} = \frac{1}{5}\)
Multiplying both sides by 8b + 3a, we get:
5b - a = \(\frac{8b + 3a}{5}\)
Expanding the right-hand side, we get:
5b - a = \(\frac{8b}{5} + \frac{3a}{5}\)
Subtracting \(\frac{8b}{5}\) from both sides, we get:
-3b + a = \(\frac{3a}{5}\)
Multiplying both sides by 5, we get:
-15b + 5a = 3a
Subtracting 3a from both sides, we get:
-15b + 2a = 0
Dividing both