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NSW Y8 Maths

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NSW Y8 Maths Equations Harder Equations

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Harder Equations Theory

For harder equations all the types: \\ brackets, variables and fractions are combined in a single equation.\\[3pt] The order of operations is brackets, fractions and moving all the variables to one side of the equation. \begin{multicols}{2} \textbf{Example 1}\\ Solve \(2\left(\dfrac{x}{3}-1\right)+x=\dfrac{x}{2}+3\)\\ \textbf{Example 1 solution}\\ \(\begin{aligned} &\frac{2 x}{3}-2+x=\frac{x}{2}+3 \\ & \text{LCD}=3 \times 2=6 \end{aligned}\)\\ \(\begin{aligned} { }^2 \cancel{6} \times \frac{2 x}{\cancel{3}}-6 \times 2+6 \times x&={ }^3 \cancel{6}\times \frac{x}{\cancel{2}}+6 \times 3\\ 4 x-12+6 x&=3 x+18 \\ 10 x-12&=3 x+18 \\ 10 x-3 x-12&=3 x-3 x+18 \\ 7 x-12&=18 \\ 7 x-12+12&=18+12 \\ 7 x&=30 \\ \frac{7 x}{7}&=\frac{30}{7} \\ x&=4 \frac{2}{7} \\ \end{aligned}\) \columnbreak \textbf{Example 2}\\ Solve \(\dfrac{5 x}{3}+\dfrac{x}{2}= \dfrac{2(x+1)}{6}\)\\ \textbf{Example 2 solution}\\ \(\begin{aligned} &\frac{5 x}{3}+\frac{x}{2}=\frac{2 x+2}{6}\\ &\text{LCD}=6 \end{aligned}\)\\ \(\begin{aligned} ^{2}\cancel{6} \times \frac{5 x}{\cancel{3}}+^{3}\cancel{6} \times \frac{x}{\cancel{2}} & =\cancel{6} \times\frac{(2 x+2)}{\cancel{6}} \\ 10 x+3 x & =2 x+2 \\ 13 x & =2 x+2 \\ 13 x-2 x & =2 x-2 x+2 \\ 11 x & =2 \\ \frac{1 x}{11} & =\frac{2}{11} \\ x & =\frac{2}{11} \end{aligned}\) \end{multicols}

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