Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{2}{7}=\frac{5}{2}x-2\)
2. \(\frac{x}{7}+\frac{4}{9}=\frac{1}{2}x-2\)
3. \(\frac{x}{4}-\frac{2}{13}=\frac{-8}{3}x-6\)
4. \(\frac{x}{4}+\frac{4}{7}=\frac{-2}{3}x+7\)
5. \(\frac{x}{3}-\frac{3}{16}=\frac{5}{4}x+4\)
6. \(\frac{x}{7}-\frac{5}{11}=\frac{1}{2}x-1\)
7. \(\frac{x}{3}+\frac{5}{11}=\frac{6}{5}x+4\)
8. \(\frac{x}{2}-\frac{4}{13}=\frac{-5}{3}x+3\)
9. \(\frac{x}{5}-\frac{2}{15}=\frac{-5}{6}x-5\)
10. \(\frac{x}{4}+\frac{3}{13}=\frac{-8}{3}x+5\)
11. \(\frac{x}{6}+\frac{2}{13}=\frac{-4}{5}x-3\)
12. \(\frac{x}{2}+\frac{2}{15}=\frac{-8}{3}x-2\)

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Reeks met breuken waarbij we ervoor kiezen om de breuken weg te werken. Neem je ZRM!

Verbetersleutel

1. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{5}{2}x-2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x-\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 35x-28 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-35x} & = & \color{red}{35x} -28 \color{blue}{-35x} \color{blue}{-4} \\\Leftrightarrow & -33x& = & -32 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-32}{-33} \\\Leftrightarrow & x = \frac{32}{33} & & \\ & V = \left\{ \frac{32}{33} \right\} & \\\end{align}\)
2. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x-\frac{252}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 63x-252 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-63x} & = & \color{red}{63x} -252 \color{blue}{-63x} \color{blue}{-56} \\\Leftrightarrow & -45x& = & -308 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-308}{-45} \\\Leftrightarrow & x = \frac{308}{45} & & \\ & V = \left\{ \frac{308}{45} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-416}{ \color{blue}{156} }x-\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & -416x-936 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+416x} & = & \color{red}{-416x} -936 \color{blue}{+416x} \color{blue}{+24} \\\Leftrightarrow & 455x& = & -912 \\\Leftrightarrow & \frac{455x}{ \color{red}{455} }& = & \frac{-912}{455} \\\Leftrightarrow & x = \frac{-912}{455} & & \\ & V = \left\{ \frac{-912}{455} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -56x+588 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+56x} & = & \color{red}{-56x} +588 \color{blue}{+56x} \color{blue}{-48} \\\Leftrightarrow & 77x& = & 540 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{540}{77} \\\Leftrightarrow & x = \frac{540}{77} & & \\ & V = \left\{ \frac{540}{77} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{5}{4}x+4 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{60}{ \color{blue}{48} }x+\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 60x+192 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-60x} & = & \color{red}{60x} +192 \color{blue}{-60x} \color{blue}{+9} \\\Leftrightarrow & -44x& = & 201 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{201}{-44} \\\Leftrightarrow & x = \frac{-201}{44} & & \\ & V = \left\{ \frac{-201}{44} \right\} & \\\end{align}\)
6. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 70 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{154}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-70& = & 77x-154 \\\Leftrightarrow & 22x \color{red}{-70} \color{blue}{+70} \color{blue}{-77x} & = & \color{red}{77x} -154 \color{blue}{-77x} \color{blue}{+70} \\\Leftrightarrow & -55x& = & -84 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-84}{-55} \\\Leftrightarrow & x = \frac{84}{55} & & \\ & V = \left\{ \frac{84}{55} \right\} & \\\end{align}\)
7. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{198}{ \color{blue}{165} }x+\frac{660}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+75& = & 198x+660 \\\Leftrightarrow & 55x \color{red}{+75} \color{blue}{-75} \color{blue}{-198x} & = & \color{red}{198x} +660 \color{blue}{-198x} \color{blue}{-75} \\\Leftrightarrow & -143x& = & 585 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{585}{-143} \\\Leftrightarrow & x = \frac{-45}{11} & & \\ & V = \left\{ \frac{-45}{11} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & -130x+234 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+130x} & = & \color{red}{-130x} +234 \color{blue}{+130x} \color{blue}{+24} \\\Leftrightarrow & 169x& = & 258 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{258}{169} \\\Leftrightarrow & x = \frac{258}{169} & & \\ & V = \left\{ \frac{258}{169} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & -25x-150 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{+25x} & = & \color{red}{-25x} -150 \color{blue}{+25x} \color{blue}{+4} \\\Leftrightarrow & 31x& = & -146 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{-146}{31} \\\Leftrightarrow & x = \frac{-146}{31} & & \\ & V = \left\{ \frac{-146}{31} \right\} & \\\end{align}\)
10. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-416}{ \color{blue}{156} }x+\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+36& = & -416x+780 \\\Leftrightarrow & 39x \color{red}{+36} \color{blue}{-36} \color{blue}{+416x} & = & \color{red}{-416x} +780 \color{blue}{+416x} \color{blue}{-36} \\\Leftrightarrow & 455x& = & 744 \\\Leftrightarrow & \frac{455x}{ \color{red}{455} }& = & \frac{744}{455} \\\Leftrightarrow & x = \frac{744}{455} & & \\ & V = \left\{ \frac{744}{455} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & -312x-1170 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+312x} & = & \color{red}{-312x} -1170 \color{blue}{+312x} \color{blue}{-60} \\\Leftrightarrow & 377x& = & -1230 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-1230}{377} \\\Leftrightarrow & x = \frac{-1230}{377} & & \\ & V = \left\{ \frac{-1230}{377} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-80}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & -80x-60 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{+80x} & = & \color{red}{-80x} -60 \color{blue}{+80x} \color{blue}{-4} \\\Leftrightarrow & 95x& = & -64 \\\Leftrightarrow & \frac{95x}{ \color{red}{95} }& = & \frac{-64}{95} \\\Leftrightarrow & x = \frac{-64}{95} & & \\ & V = \left\{ \frac{-64}{95} \right\} & \\\end{align}\)