$a = \frac{3 - b}{b + 4} + 2$ Find a when b = -6. Answer a = ... Rearrange the formula to make b the subject. Answer b = ... Solve these simultaneous equations. $8x + 5y = 9$ $6x - 2y = 47$ You must show your working. Answer x = ... y = ... Write as a single fraction in its simplest form $\frac{x}{2x-1} - \frac{3}{x+4}$. Answer ... Solve the equation $\dfrac{11}{x-3} = 3x - 1$. Answer x = ... or x = ...

1. Substitute $b=-6$ into the formula and evaluate $a$.

$a=\frac{3-(-6)}{-6+4}+2$
$a=\frac{9}{-2}+2=-\frac{9}{2}+\frac{4}{2}=-\frac{5}{2}$

1. Isolate the algebraic fraction involving $b$.

$a=\dfrac{3-b}{b+4}+2$

$a-2=\dfrac{3-b}{b+4}$

2. Clear the denominator and expand.

$(a-2)(b+4)=3-b$

$(a-2)b+4(a-2)=3-b$

3. Collect the terms in $b$ and divide by the coefficient of $b$.

$(a-2)b+b=3-4(a-2)$

$b[(a-2)+1]=3-4(a-2)$

$b(a-1)=3-4(a-2)$

$b=\dfrac{3-4(a-2)}{a-1}$

1. Make the coefficients of $y$ equal in magnitude.

Multiply $(1)$ by $2$ and $(2)$ by $5$:

2. Eliminate $y$ and solve for $x$.

Add $(3)$ and $(4)$:
$46x=253$
$x=\frac{253}{46}$

3. Substitute the value of $x$ into one original equation to find $y$.

Using $6x-2y=47$,
$6\left(\frac{253}{46}\right)-2y=47$
$\frac{759}{23}-2y=47$
$-2y=47-\frac{759}{23}=\frac{1081-759}{23}=\frac{322}{23}$
$y=-\frac{161}{23}$

1. Use a common denominator to write the two fractions over one denominator.

The common denominator is $(2x-1)(x+4)$.

$\frac{x}{2x-1}-\frac{3}{x+4}=\frac{x(x+4)-3(2x-1)}{(2x-1)(x+4)}$

2. Expand and simplify the numerator.

$\frac{x(x+4)-3(2x-1)}{(2x-1)(x+4)}=\frac{x^2+4x-6x+3}{(2x-1)(x+4)}$

$=\frac{x^2-2x+3}{(2x-1)(x+4)}$

1. Clear the denominator by multiplying both sides by $x-3$.

$\dfrac{11}{x-3}=3x-1$

Since $x\ne 3$,

$11=(3x-1)(x-3)$

2. Expand and rearrange into standard quadratic form.

$11=3x^2-9x-x+3$

$11=3x^2-10x+3$

$3x^2-10x-8=0$

3. Factorise and solve the quadratic equation.

$3x^2-10x-8=0$

$(3x+2)(x-4)=0$

$3x+2=0$ or $x-4=0$

$x=-\dfrac{2}{3}$ or $x=4$