Question

simplify and solve the following equations :
$a) \: \: 2( \frac{2}{3}x - 9) + \frac{5}{2} x - 4 = - 1$
$b) \: \: x = \frac{5}{4} (x + 9)$
$c) \: \: - 5.5c(c - 2.25) = - 1.1c(5c - 8.7) - 1.5$

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Answer 1

Answer:

Use grouping to factorise the numerator and take out the common factor \(ax\) in the denominator\[\frac{\left(ax - ab\right) + \left(x - b\right)}{a{x}^{2} - abx} = \frac{a\left(x - b\right) + \left(x - b\right)}{ax\left(x - b\right)}\]

Take out common factor \(\left(x-b\right)\) in the numerator\[=\frac{\left(x - b\right)\left(a + 1\right)}{ax\left(x - b\right)}\]

Cancel the common factor in the numerator and the denominator to give the final answer\[= \frac{a + 1}{ax}\]

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Answer 2

Step-by-step explanation:

$\bf \underline{Solution-}$

$\bf{Q.(i )} \: \: 2 \bigg( \frac{2}{3} x - 9 \bigg) + \frac{5}{2} x - 4 = - 1$

$\rm\longrightarrow \: 2 \times \bigg( \frac{2}{3} \bigg)x - 18 + \frac{5}{2} x - 4 = - 1$

$\rm\longrightarrow \: \frac{2 \times 2}{3} x - 18 + \frac{5}{2} x - 4 = - 1$

$\rm\longrightarrow \: \frac{4}{3} x - 18 + \frac{5}{2}x - 4 = - 1$

Combine (4/3)x and (5/2)x to get (23/6)x.

$\rm\longrightarrow \: \frac{23}{6} x - 18 - 4 = - 1$

$\rm\longrightarrow \: \frac{23}{6} x - 22 = - 1$

$\rm\longrightarrow \: \frac{23}{6} x = - 1 + 22$

$\rm\longrightarrow \:\frac{23}{6} x = 21$

Multiply both sides by 6/23, the reciprocal of 23/6.

$\rm\longrightarrow \:x = 21 \times \bigg( \frac{6}{23} \bigg)$

Express 21*(6/23) as a single fraction.

$\rm\longrightarrow \:x = \frac{21 \times 6}{23}$

$\rm\longrightarrow \:x = \frac{126}{23} = 5 \frac{11}{23} \bf \: ans.$

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$\bf{Q.(ii)} \: \: x = \frac{5}{4} (x + 9)$

$\rm\longrightarrow \:x = \frac{5}{4} x + \frac{5}{4} \times 9$

Express (5/4)*9 as a single fraction.

$\rm\longrightarrow \:x = \frac{5}{4} x + \frac{5 \times 9}{4}$

$\rm\longrightarrow \:x = \frac{5}{4} x + \frac{45}{4}$

$\rm\longrightarrow \:x - \frac{5}{4} x = \frac{45}{4}$

$\rm\longrightarrow \: \frac{ - 1}{4} x = \frac{45}{4}$

$\rm\longrightarrow \:x = \frac{45}{4} ( - 4)$

$\rm\longrightarrow \:x = \frac{45( - 4)}{4}$

$\rm\longrightarrow \:x = \frac{ - 180}{4}$

$\rm\longrightarrow \:x = - 45 \bf \: ans.$

$\underline{ \bf \therefore \: value \: of \: x \: is \: - 45.}$

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$\bf{Q.(iii)} \: \: - 5.5c(c.2.25) = - 1.1c(5c - 8.7) - 1.5$

$\rm \longrightarrow \: - 5.5 {c}^{2} + 12.375c = - 1.1c(5c - 8.7) - 1.5$

$\rm \longrightarrow \: - 5.5 {c}^{2} + 12.375c = - 5.5 {c}^{2} + 9.57c - 1.5$

$\rm \longrightarrow \: \cancel{- 5.5 {c}^{2}} + 12.375c \cancel{+ 5.5 {c}^{2}} = 9.57c - 1.5$

$\rm \longrightarrow \: 12.375c = 9.57c - 1.5$

$\rm \longrightarrow \: 12.375c - 9.57c = - 1.5$

$\rm \longrightarrow \: 2.805c = - 1.5$

$\rm \longrightarrow \: c = \frac{1.5}{2.805}$

Expand -1.5/2.805 by multiplying both numerator and the denominator by 1000.

$\rm \longrightarrow \: c = \frac{ - 1500}{2805}$

Reduce the fraction -1500/2805 to lowest terms by extracting and cancaling out 15.

$\rm \longrightarrow \: c = \frac{100}{187} = 0.534759358$

$\underline{\bf{Hence, the \: reqrd \: value \: of \: c \: is \: 0.534759358}}$