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Solve linear equations with unkowns on both sides. Including brackets and fractions.

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Part a)

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Given $\\simplify{{l}({m}w-{n}) = {p}w+{q}}$, we can expand the brackets, get all the $w$'s on the left hand side and all the numbers on the right hand side, and then divide both sides by the coefficient of $w$ to get $w$ by itself.

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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\simplify{{l}({m}w-{n})}$$=$$\\simplify{{p}w+{q}}$ 
 
$\\simplify{{l*m}w+{n*l}}$$=$$\\simplify{{p}w+{q}}$
 
$\\simplify[!cancelTerms,unitFactor]{{l*m}w-{n*l}-{p}w}$$=$$\\simplify[!cancelTerms,unitFactor]{{p}w+{q}-{p}w}$
 
$\\simplify{{l*m-p}w-{n*l}}$$=$$\\var{q}$
 
$\\var{l*m-p}w-\\var{n*l}+\\var{n*l}$$=$$\\var{q}+\\var{n*l}$
 
$\\var{l*m-p}w$$=$$\\var{q+n*l}$
 
$\\displaystyle{\\frac{\\var{l*m-p}w}{\\var{l*m-p}}}$$=$$\\displaystyle{\\frac{\\var{q+n*l}}{\\var{l*m-p}}}$
 
$w$$=$$\\displaystyle{\\simplify{{q+n*l}/{l*m-p}}} = \\var{precround(ansA,1)} \\text{ to 1 dp}$
\n

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Part b)

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Given $\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}=\\var{g}$, we can multiply both sides by $(y-\\var{f})$ to get rid of the fraction, get all the $y$'s on one side and the numbers on the other side, and then divide both sides by the coefficient of $y$ to get $y$ by itself.

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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}$$=$$\\var{g}$ 
 
$\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}\\times(y-\\var{f})$$=$$\\var{g}\\times (y-\\var{f})$
 
$\\var{d}y$$=$$\\simplify[unitFactor]{{g}y+{-g*f}}$
 
$\\simplify[!cancelTerms,unitFactor]{{d}y+{-g}y}$ $=$$\\simplify[!cancelTerms,unitFactor]{{g}y+{-g*f}+{-g}y}$
 
$\\var{d-g}y$$=$$\\var{-g*f}$
 
$\\displaystyle{\\frac{\\var{d-g}y}{\\var{d-g}}}$$=$$\\displaystyle{\\frac{\\var{-g*f}}{\\var{d-g}}}$
 
 $y$$=$$\\displaystyle{\\simplify{{-g*f}/{d-g}}}= \\var{precround(ansB,1)}\\text{ to 1 dp}$
\n

part c)

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Given $\\displaystyle{\\frac{x+\\var{add}}{\\var{denom1}}+\\frac{x}{\\var{denom2}}=\\var{right}}$, we can multiply both sides by $\\var{denom1}$ and by $\\var{denom2}$ to get rid of the fractions, get all the $x$'s on one side and the numbers on the other side, and then divide both sides by the coefficient of $x$ to get $x$ by itself.

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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\displaystyle{\\frac{x+\\var{add}}{\\var{denom1}}+\\frac{x}{\\var{denom2}}}$$=$$\\var{right}$ 
 
$\\displaystyle{\\left(\\frac{x+\\var{add}}{\\var{denom1}}\\right)\\times\\var{denom1}+\\left(\\frac{x}{\\var{denom2}}\\right)\\times\\var{denom1}}$$=$$\\var{right}\\times \\var{denom1}$(multiply all terms by $\\var{denom1}$)
 
$\\displaystyle{x+\\var{add}+\\frac{\\var{denom1}x}{\\var{denom2}}}$$=$$\\var{r1}$
 
$\\displaystyle{(x+\\var{add})\\times\\var{denom2}+\\left(\\frac{\\var{denom1}x}{\\var{denom2}}\\right)\\times\\var{denom2}}$ $=$$\\var{r1}\\times\\var{denom2}$(multiply all terms by $\\var{denom2}$)
 
$\\displaystyle{\\var{denom2}x+\\var{a2}+\\var{denom1}x}$$=$$\\var{r12}$
 
$\\var{sumdeno}x+\\var{a2}$$=$$\\var{r12}$(collect like terms)
 
 $\\var{sumdeno}x$$=$$\\var{r12}-\\var{a2}$(collect like terms)
 
$\\var{sumdeno}x$$=$$\\var{top}$
 
$x$$=$$\\displaystyle{\\simplify{{top}/({sumdeno})}}= \\var{precround(ansD,1)} \\text{ to 1 dp}$(divide by the coefficient of $x$)
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Solve  $\\simplify{{l}({m}w-{n}) = {p}w+{q}}$

\n

$w=$ [[0]]

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Given $\\displaystyle{\\frac{\\var{d}y}{y-\\var{f}}}=\\var{g}$,  

\n

$y=$ [[0]].

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Solve $\\displaystyle{\\frac{x+\\var{add}}{\\var{denom1}}+\\frac{x}{\\var{denom2}}=\\var{right}}$.

\n

$x=$ [[0]]

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