a point q on a segment with endpoints a (2, −1) and c (4, 2) partitions the segment in a 4: 1 ratio. find q. you must show all work to receive credit.

point q: (3.6, 1.4)

step-by-step explanation:

when you need to find a point between 2 points, you can work with "x" and with "y" separately:

point a: (2, -1)

point c: (4, 2)

distance in x: (4-2) = 2

distance in y: ()) = 3

the point q partitions the segment in 4: 1, from that you can divide the segment in 5 parts.

each part in x = 2/5 = 0.4

each part in y = 3/5 = 0.6

i will assume that the 4 parts are from a to q (because it isnt specified in the problem)

q: point a + 4 parts:

point q: (2, -1) +(0.4*4, 0.6*4) = (3.6, 1,4)

Since we want to solve for the variable x, we want to isolate x

a²x + (a - 1) = (a + 1)x ⇒ Distribute x to (a+1). Also, remove parentheses

a²x + a - 1 = ax + x ⇒ Subtract a from both sides

a²x - 1 = ax + x - a ⇒ Add 1 to both sides

a²x = ax + x - a + 1 ⇒ Subtract (ax + x) from both sides

a²x - (ax + x)= ax + x - a + 1 - (ax+x) ⇒ Simplify. Remember that multiplying positive by negative = negative

a²x - ax - x = ax + x - a + 1 - ax - x ⇒ Simplify

a²x - ax - x = -a + 1 ⇒ Factor out the x from a²x - ax - x

x(a² - a - 1) = -a + 1 ⇒ Divide both sides by (a² - a - 1)

x = (-a + 1) / (a² - a - 1)

However, we need to make sure that the denominator does not equal 0. Therefore, you set the denominator = 0 (just use the quadratic formula for this), and it gives that the denominator =0 when a = (1+√5)/2 AND (1-√5)/2

Therefore, the final answer is

x = (-a + 1) / (a² - a - 1) given that a ≠ (1+√5)/2, a ≠ (1-√5)/2