a. The sum of three numbers is 81. The second number is twice the first.

Explanation

a. Let the numbers be p, r and q respectively

Given:

p + r + q = 81 --- (i)

r = 2p --- (ii)

q = r + 6 --- (iii)

Substitute (2p) for r in equation (iii)

q = 2p + 6 --- (iv)

Substitute (2p) for r and (2p + 6) for q in equation (i)

⇒ p + (2p) + (2p + 6) = 81

⇒ 5p + 6 = 81

⇒ 5p = 81 - 6

⇒ 5p = 75

p = \(\frac{75}{5}\) = 15.

From equation (ii)

⇒ r = 2(15) = 30

From equation (iv)

⇒ q = 2(15) + 6

⇒ q = 30 + 6 = 36

∴ The numbers are 15, 30 and 36

b. \(m_1 = \frac{ y_2 - y_1}{ x_2 - x_1}\)

Given points are P( 3, 5) and Q( -5, 7) = P(\( x_1 , y_1)\) , Q(\( x_2 , y_2)\)

the slope of line PQ = \((\frac{ 7 - 5}{ -5 - 3}) = \frac{2}{-8} = -\frac{1}{4}\)

Given that R( x, y) is the midpoint of the line formed by the points P( 3, 5) and Q( -5, 7)

R(x, y) = \(\frac{3 + (-5)}{2}, \frac{5 + 7}{2} = \frac{ 3 - 5}{2} , \frac{ 5+7}{2}\)

= -\(\frac{2}{2} , \frac{12}{2}\) = (-1, 6)

but the line passes through the point R( -1, 6) and is perpendicular to LINE PQ

\(m_2 = \frac{1}{m_1} = -\frac{1}{\frac{-1}{4}}\) = 4

Equation of the line can be obtained using \(y - y_1 = m_2( x - x_1)\)

y - 6 = 4( x - (-1)) = y - 6 = 4( x + 1 )

y - 6 = 4x + 4

y = 4x + 4 + 6

Therefore, y = 4x + 10