The problem has two parts. (a) Simplify the expression $5(6 - ab) + 2(-7 + 3ab)$. (b) Given the equation of a straight line $3x - 2y - 6 = 0$, find the gradient and the y-intercept of the line.

1. Problem Description

The problem has two parts.

(a) Simplify the expression $5(6 - ab) + 2(-7 + 3ab)$.

(b) Given the equation of a straight line $3x - 2y - 6 = 0$, find the gradient and the y-intercept of the line.

2. Solution Steps

(a) To simplify the expression $5(6 - ab) + 2(-7 + 3ab)$, we first distribute the numbers outside the parentheses:

$5(6 - ab) = 5 \cdot 6 - 5 \cdot ab = 30 - 5ab$

$2(-7 + 3ab) = 2 \cdot (-7) + 2 \cdot 3ab = -14 + 6ab$

Then, we add the two results:

$30 - 5ab + (-14 + 6ab) = 30 - 5ab - 14 + 6ab = (30 - 14) + (-5ab + 6ab) = 16 + ab$

(b) The equation of a straight line is given by $3x - 2y - 6 = 0$. We want to find the gradient and the y-intercept. First, we rewrite the equation in the slope-intercept form, which is $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept.

$3x - 2y - 6 = 0$

$-2y = -3x + 6$

$y = \frac{-3x + 6}{-2}$

$y = \frac{3}{2}x - 3$

Therefore, the gradient of the line is $\frac{3}{2}$ and the y-intercept is -

3.

3. Final Answer

(a) $16 + ab$

(b) (i) Gradient: $\frac{3}{2}$

(ii) y-intercept: $-3$