Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{3}{4}x+2 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 21 }{ \color{blue}{112} })& = & (\frac{84}{ \color{blue}{112} }x+\frac{224}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-21& = & 84x+224 \\\Leftrightarrow & 16x \color{red}{-21} \color{blue}{+21} \color{blue}{-84x} & = & \color{red}{84x} +224 \color{blue}{-84x} \color{blue}{+21} \\\Leftrightarrow & -68x& = & 245 \\\Leftrightarrow & \frac{-68x}{ \color{red}{-68} }& = & \frac{245}{-68} \\\Leftrightarrow & x = \frac{-245}{68} & & \\ & V = \left\{ \frac{-245}{68} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 36x+60 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-36x} & = & \color{red}{36x} +60 \color{blue}{-36x} \color{blue}{+8} \\\Leftrightarrow & -31x& = & 68 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{68}{-31} \\\Leftrightarrow & x = \frac{-68}{31} & & \\ & V = \left\{ \frac{-68}{31} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-7}{4}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+60& = & -245x-980 \\\Leftrightarrow & 28x \color{red}{+60} \color{blue}{-60} \color{blue}{+245x} & = & \color{red}{-245x} -980 \color{blue}{+245x} \color{blue}{-60} \\\Leftrightarrow & 273x& = & -1040 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{-1040}{273} \\\Leftrightarrow & x = \frac{-80}{21} & & \\ & V = \left\{ \frac{-80}{21} \right\} & \\\end{align}\)
4. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{4}{3}x-1 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{8}{ \color{blue}{6} }x-\frac{6}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 8x-6 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-8x} & = & \color{red}{8x} -6 \color{blue}{-8x} \color{blue}{-5} \\\Leftrightarrow & -5x& = & -11 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-11}{-5} \\\Leftrightarrow & x = \frac{11}{5} & & \\ & V = \left\{ \frac{11}{5} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 154x+66 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-154x} & = & \color{red}{154x} +66 \color{blue}{-154x} \color{blue}{+30} \\\Leftrightarrow & -121x& = & 96 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{96}{-121} \\\Leftrightarrow & x = \frac{-96}{121} & & \\ & V = \left\{ \frac{-96}{121} \right\} & \\\end{align}\)
6. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 65x-130 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-65x} & = & \color{red}{65x} -130 \color{blue}{-65x} \color{blue}{-40} \\\Leftrightarrow & -39x& = & -170 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-170}{-39} \\\Leftrightarrow & x = \frac{170}{39} & & \\ & V = \left\{ \frac{170}{39} \right\} & \\\end{align}\)
7. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 60 }{ \color{blue}{195} })& = & (\frac{-78}{ \color{blue}{195} }x+\frac{390}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+60& = & -78x+390 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+78x} & = & \color{red}{-78x} +390 \color{blue}{+78x} \color{blue}{-60} \\\Leftrightarrow & 143x& = & 330 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{330}{143} \\\Leftrightarrow & x = \frac{30}{13} & & \\ & V = \left\{ \frac{30}{13} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & -12x-150 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+12x} & = & \color{red}{-12x} -150 \color{blue}{+12x} \color{blue}{-4} \\\Leftrightarrow & 17x& = & -154 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-154}{17} \\\Leftrightarrow & x = \frac{-154}{17} & & \\ & V = \left\{ \frac{-154}{17} \right\} & \\\end{align}\)
9. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{7}{3}x-2 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 84 }{ \color{blue}{231} })& = & (\frac{539}{ \color{blue}{231} }x-\frac{462}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+84& = & 539x-462 \\\Leftrightarrow & 33x \color{red}{+84} \color{blue}{-84} \color{blue}{-539x} & = & \color{red}{539x} -462 \color{blue}{-539x} \color{blue}{-84} \\\Leftrightarrow & -506x& = & -546 \\\Leftrightarrow & \frac{-506x}{ \color{red}{-506} }& = & \frac{-546}{-506} \\\Leftrightarrow & x = \frac{273}{253} & & \\ & V = \left\{ \frac{273}{253} \right\} & \\\end{align}\)
10. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 70 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{1078}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-70& = & 77x+1078 \\\Leftrightarrow & 22x \color{red}{-70} \color{blue}{+70} \color{blue}{-77x} & = & \color{red}{77x} +1078 \color{blue}{-77x} \color{blue}{+70} \\\Leftrightarrow & -55x& = & 1148 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{1148}{-55} \\\Leftrightarrow & x = \frac{-1148}{55} & & \\ & V = \left\{ \frac{-1148}{55} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -84x+420 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} +420 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & 360 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{360}{119} \\\Leftrightarrow & x = \frac{360}{119} & & \\ & V = \left\{ \frac{360}{119} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{3}{2}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{120}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & 120x+400 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-120x} & = & \color{red}{120x} +400 \color{blue}{-120x} \color{blue}{+15} \\\Leftrightarrow & -104x& = & 415 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{415}{-104} \\\Leftrightarrow & x = \frac{-415}{104} & & \\ & V = \left\{ \frac{-415}{104} \right\} & \\\end{align}\)