a) Find the gradient and y-intercept of the line given by the equation $x + 2y = 14$. b) Find the equation of the line that passes through the points $(-3, 1)$ and $(2, -14)$. c) Determine the partial fraction decomposition of the expression $\frac{17x - 53}{x^2 - 2x - 15}$.

1. Problem Description

a) Find the gradient and y-intercept of the line given by the equation $x + 2y = 14$.

b) Find the equation of the line that passes through the points $(-3, 1)$ and $(2, -14)$.

c) Determine the partial fraction decomposition of the expression $\frac{17x - 53}{x^2 - 2x - 15}$.

2. Solution Steps

a) To find the gradient and y-intercept of the line $x + 2y = 14$, we need to rewrite the equation in the slope-intercept form, which is $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept.

$x + 2y = 14$

$2y = -x + 14$

$y = -\frac{1}{2}x + 7$

Therefore, the gradient is $-\frac{1}{2}$ and the y-intercept is $7$.

b) To find the equation of the line passing through the points $(-3, 1)$ and $(2, -14)$, we first need to find the gradient $m$.

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates of the given points:

$m = \frac{-14 - 1}{2 - (-3)} = \frac{-15}{5} = -3$

Now, we can use the point-slope form of a linear equation:

$y - y_1 = m(x - x_1)$

Using the point $(-3, 1)$ and the gradient $m = -3$:

$y - 1 = -3(x - (-3))$

$y - 1 = -3(x + 3)$

$y - 1 = -3x - 9$

$y = -3x - 8$

So, the equation of the line is $y = -3x - 8$.

c) To determine the partial fraction decomposition of the expression $\frac{17x - 53}{x^2 - 2x - 15}$, we first need to factor the denominator.

$x^2 - 2x - 15 = (x - 5)(x + 3)$

Now, we can express the given fraction as a sum of two fractions with the factored denominators:

$\frac{17x - 53}{(x - 5)(x + 3)} = \frac{A}{x - 5} + \frac{B}{x + 3}$

Multiplying both sides by $(x - 5)(x + 3)$, we get:

$17x - 53 = A(x + 3) + B(x - 5)$

Now, we can solve for $A$ and $B$ by choosing appropriate values for $x$.

Let $x = 5$:

$17(5) - 53 = A(5 + 3) + B(5 - 5)$

$85 - 53 = 8A + 0$

$32 = 8A$

$A = 4$

Let $x = -3$:

$17(-3) - 53 = A(-3 + 3) + B(-3 - 5)$

$-51 - 53 = 0 + B(-8)$

$-104 = -8B$

$B = 13$

So, the partial fraction decomposition is:

$\frac{17x - 53}{x^2 - 2x - 15} = \frac{4}{x - 5} + \frac{13}{x + 3}$

3. Final Answer

a) Gradient: $-\frac{1}{2}$, y-intercept: $7$

b) $y = -3x - 8$

c) $\frac{4}{x - 5} + \frac{13}{x + 3}$