Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{5}{13}=\frac{5}{3}x-8\)
2. \(\frac{x}{7}-\frac{4}{13}=\frac{-3}{4}x-1\)
3. \(\frac{x}{2}-\frac{3}{7}=\frac{7}{3}x-8\)
4. \(\frac{x}{4}+\frac{4}{7}=\frac{-4}{5}x-8\)
5. \(\frac{x}{3}-\frac{5}{11}=\frac{1}{2}x+6\)
6. \(\frac{x}{4}+\frac{5}{9}=\frac{-2}{3}x-5\)
7. \(\frac{x}{6}+\frac{5}{7}=\frac{-4}{5}x+1\)
8. \(\frac{x}{7}-\frac{4}{7}=\frac{-7}{4}x-5\)
9. \(\frac{x}{3}-\frac{5}{16}=\frac{-7}{4}x-7\)
10. \(\frac{x}{2}+\frac{5}{9}=\frac{-2}{3}x+4\)
11. \(\frac{x}{2}+\frac{3}{14}=\frac{-2}{5}x+3\)
12. \(\frac{x}{4}-\frac{3}{10}=\frac{2}{3}x-4\)

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

Verbetersleutel

1. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{13}& = & \frac{5}{3}x-8 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 105 }{ \color{blue}{273} })& = & (\frac{455}{ \color{blue}{273} }x-\frac{2184}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+105& = & 455x-2184 \\\Leftrightarrow & 39x \color{red}{+105} \color{blue}{-105} \color{blue}{-455x} & = & \color{red}{455x} -2184 \color{blue}{-455x} \color{blue}{-105} \\\Leftrightarrow & -416x& = & -2289 \\\Leftrightarrow & \frac{-416x}{ \color{red}{-416} }& = & \frac{-2289}{-416} \\\Leftrightarrow & x = \frac{2289}{416} & & \\ & V = \left\{ \frac{2289}{416} \right\} & \\\end{align}\)
2. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-3}{4}x-1 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 112 }{ \color{blue}{364} })& = & (\frac{-273}{ \color{blue}{364} }x-\frac{364}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-112& = & -273x-364 \\\Leftrightarrow & 52x \color{red}{-112} \color{blue}{+112} \color{blue}{+273x} & = & \color{red}{-273x} -364 \color{blue}{+273x} \color{blue}{+112} \\\Leftrightarrow & 325x& = & -252 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{-252}{325} \\\Leftrightarrow & x = \frac{-252}{325} & & \\ & V = \left\{ \frac{-252}{325} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 98x-336 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-98x} & = & \color{red}{98x} -336 \color{blue}{-98x} \color{blue}{+18} \\\Leftrightarrow & -77x& = & -318 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-318}{-77} \\\Leftrightarrow & x = \frac{318}{77} & & \\ & V = \left\{ \frac{318}{77} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x-\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -112x-1120 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+112x} & = & \color{red}{-112x} -1120 \color{blue}{+112x} \color{blue}{-80} \\\Leftrightarrow & 147x& = & -1200 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{-1200}{147} \\\Leftrightarrow & x = \frac{-400}{49} & & \\ & V = \left\{ \frac{-400}{49} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-30& = & 33x+396 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-33x} & = & \color{red}{33x} +396 \color{blue}{-33x} \color{blue}{+30} \\\Leftrightarrow & -11x& = & 426 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{426}{-11} \\\Leftrightarrow & x = \frac{-426}{11} & & \\ & V = \left\{ \frac{-426}{11} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{9}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 20 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x-\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+20& = & -24x-180 \\\Leftrightarrow & 9x \color{red}{+20} \color{blue}{-20} \color{blue}{+24x} & = & \color{red}{-24x} -180 \color{blue}{+24x} \color{blue}{-20} \\\Leftrightarrow & 33x& = & -200 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-200}{33} \\\Leftrightarrow & x = \frac{-200}{33} & & \\ & V = \left\{ \frac{-200}{33} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & -168x+210 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{+168x} & = & \color{red}{-168x} +210 \color{blue}{+168x} \color{blue}{-150} \\\Leftrightarrow & 203x& = & 60 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{60}{203} \\\Leftrightarrow & x = \frac{60}{203} & & \\ & V = \left\{ \frac{60}{203} \right\} & \\\end{align}\)
8. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-7}{4}x-5 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 16 }{ \color{blue}{28} })& = & (\frac{-49}{ \color{blue}{28} }x-\frac{140}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-16& = & -49x-140 \\\Leftrightarrow & 4x \color{red}{-16} \color{blue}{+16} \color{blue}{+49x} & = & \color{red}{-49x} -140 \color{blue}{+49x} \color{blue}{+16} \\\Leftrightarrow & 53x& = & -124 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{-124}{53} \\\Leftrightarrow & x = \frac{-124}{53} & & \\ & V = \left\{ \frac{-124}{53} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{-7}{4}x-7 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-84}{ \color{blue}{48} }x-\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & -84x-336 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{+84x} & = & \color{red}{-84x} -336 \color{blue}{+84x} \color{blue}{+15} \\\Leftrightarrow & 100x& = & -321 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{-321}{100} \\\Leftrightarrow & x = \frac{-321}{100} & & \\ & V = \left\{ \frac{-321}{100} \right\} & \\\end{align}\)
10. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x+\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & -12x+72 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{+12x} & = & \color{red}{-12x} +72 \color{blue}{+12x} \color{blue}{-10} \\\Leftrightarrow & 21x& = & 62 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{62}{21} \\\Leftrightarrow & x = \frac{62}{21} & & \\ & V = \left\{ \frac{62}{21} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{14}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 15 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x+\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+15& = & -28x+210 \\\Leftrightarrow & 35x \color{red}{+15} \color{blue}{-15} \color{blue}{+28x} & = & \color{red}{-28x} +210 \color{blue}{+28x} \color{blue}{-15} \\\Leftrightarrow & 63x& = & 195 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{195}{63} \\\Leftrightarrow & x = \frac{65}{21} & & \\ & V = \left\{ \frac{65}{21} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{2}{3}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{40}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 40x-240 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-40x} & = & \color{red}{40x} -240 \color{blue}{-40x} \color{blue}{+18} \\\Leftrightarrow & -25x& = & -222 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-222}{-25} \\\Leftrightarrow & x = \frac{222}{25} & & \\ & V = \left\{ \frac{222}{25} \right\} & \\\end{align}\)