Resolve as equações
Equações do 1.º grau: Matematicamente Falando 8 - Parte 2 Pág. 56 Ex. 1
Enunciado
Resolve as equações:
- $\frac{y}{2}-\frac{2y+1}{3}=0$
- $b-(2b-4)=\frac{b}{5}$
- $\frac{5(x+2)}{2}-\frac{x}{5}=5$
- $\frac{4d-3}{8}-\frac{d}{2}=0$
- $\frac{m+3}{6}-\frac{2(m-1)}{3}=\frac{1}{9}$
Resolução
- Ora,
\[\begin{array}{*{35}{l}}
\frac{y}{\underset{(3)}{\mathop{2}}\,}-\frac{2y+1}{\underset{(2)}{\mathop{3}}\,}=\underset{(6)}{\mathop{0}}\, & \Leftrightarrow & 3y-4y-2=0 \\
{} & \Leftrightarrow & -y=2 \\
{} & \Leftrightarrow & y=-2 \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
b-(2b-4)=\frac{b}{5} & \Leftrightarrow & \underset{(5)}{\mathop{b}}\,-\underset{(5)}{\mathop{2b}}\,+\underset{(5)}{\mathop{4}}\,=\frac{b}{\underset{(1)}{\mathop{5}}\,} \\
{} & \Leftrightarrow & 5b-10b+20=b \\
{} & \Leftrightarrow & -6b=-20 \\
{} & \Leftrightarrow & b=\frac{20}{6} \\
{} & \Leftrightarrow & b=\frac{10}{3} \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
\frac{5(x+2)}{2}-\frac{x}{5}=5 & \Leftrightarrow & \frac{5x+10}{\underset{(5)}{\mathop{2}}\,}-\frac{x}{\underset{(2)}{\mathop{5}}\,}=\underset{(10)}{\mathop{5}}\, \\
{} & \Leftrightarrow & 25x+50-2x=50 \\
{} & \Leftrightarrow & 23x=0 \\
{} & \Leftrightarrow & x=0 \\
\end{array}\] - Ora,
\[\begin{array}{*{35}{l}}
\frac{4d-3}{\underset{(1)}{\mathop{8}}\,}-\frac{d}{\underset{(4)}{\mathop{2}}\,}=\underset{(8)}{\mathop{0}}\, & \Leftrightarrow & 4d-3-4d=0 \\
{} & \Leftrightarrow & 0d=3 \\
\end{array}\]
A equação é impossível. - Ora,
\[\begin{array}{*{35}{l}}
\frac{m+3}{6}-\frac{2(m-1)}{3}=\frac{1}{9} & \Leftrightarrow & \frac{m+3}{\underset{(3)}{\mathop{6}}\,}-\frac{2m-2}{\underset{(6)}{\mathop{3}}\,}=\frac{1}{\underset{(2)}{\mathop{9}}\,} \\
{} & \Leftrightarrow & 3m+9-12m+12=2 \\
{} & \Leftrightarrow & -9m=-19 \\
{} & \Leftrightarrow & m=\frac{19}{9} \\
\end{array}\]
