Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{4}{7}=\frac{1}{3}x+8\)
2. \(\frac{x}{6}+\frac{2}{7}=\frac{6}{5}x+3\)
3. \(\frac{x}{7}+\frac{3}{7}=\frac{1}{6}x-6\)
4. \(\frac{x}{5}-\frac{3}{14}=\frac{-5}{6}x+6\)
5. \(\frac{x}{7}-\frac{2}{15}=\frac{4}{3}x+2\)
6. \(\frac{x}{6}+\frac{4}{9}=\frac{-7}{5}x+4\)
7. \(\frac{x}{5}-\frac{5}{12}=\frac{4}{3}x-1\)
8. \(\frac{x}{3}-\frac{4}{7}=\frac{5}{2}x-8\)
9. \(\frac{x}{5}-\frac{4}{13}=\frac{-5}{6}x+5\)
10. \(\frac{x}{6}+\frac{5}{13}=\frac{-7}{5}x+7\)
11. \(\frac{x}{4}-\frac{3}{10}=\frac{4}{3}x-2\)
12. \(\frac{x}{7}+\frac{2}{11}=\frac{-7}{4}x+2\)

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 14x+336 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-14x} & = & \color{red}{14x} +336 \color{blue}{-14x} \color{blue}{-24} \\\Leftrightarrow & 7x& = & 312 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{312}{7} \\\Leftrightarrow & x = \frac{312}{7} & & \\ & V = \left\{ \frac{312}{7} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 252x+630 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-252x} & = & \color{red}{252x} +630 \color{blue}{-252x} \color{blue}{-60} \\\Leftrightarrow & -217x& = & 570 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{570}{-217} \\\Leftrightarrow & x = \frac{-570}{217} & & \\ & V = \left\{ \frac{-570}{217} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+18& = & 7x-252 \\\Leftrightarrow & 6x \color{red}{+18} \color{blue}{-18} \color{blue}{-7x} & = & \color{red}{7x} -252 \color{blue}{-7x} \color{blue}{-18} \\\Leftrightarrow & -x& = & -270 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-270}{-1} \\\Leftrightarrow & x = 270 & & \\ & V = \left\{ 270 \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-45& = & -175x+1260 \\\Leftrightarrow & 42x \color{red}{-45} \color{blue}{+45} \color{blue}{+175x} & = & \color{red}{-175x} +1260 \color{blue}{+175x} \color{blue}{+45} \\\Leftrightarrow & 217x& = & 1305 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{1305}{217} \\\Leftrightarrow & x = \frac{1305}{217} & & \\ & V = \left\{ \frac{1305}{217} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{140}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & 140x+210 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{-140x} & = & \color{red}{140x} +210 \color{blue}{-140x} \color{blue}{+14} \\\Leftrightarrow & -125x& = & 224 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{224}{-125} \\\Leftrightarrow & x = \frac{-224}{125} & & \\ & V = \left\{ \frac{-224}{125} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -126x+360 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+126x} & = & \color{red}{-126x} +360 \color{blue}{+126x} \color{blue}{-40} \\\Leftrightarrow & 141x& = & 320 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{320}{141} \\\Leftrightarrow & x = \frac{320}{141} & & \\ & V = \left\{ \frac{320}{141} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{4}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & 80x-60 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{-80x} & = & \color{red}{80x} -60 \color{blue}{-80x} \color{blue}{+25} \\\Leftrightarrow & -68x& = & -35 \\\Leftrightarrow & \frac{-68x}{ \color{red}{-68} }& = & \frac{-35}{-68} \\\Leftrightarrow & x = \frac{35}{68} & & \\ & V = \left\{ \frac{35}{68} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 105x-336 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-105x} & = & \color{red}{105x} -336 \color{blue}{-105x} \color{blue}{+24} \\\Leftrightarrow & -91x& = & -312 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-312}{-91} \\\Leftrightarrow & x = \frac{24}{7} & & \\ & V = \left\{ \frac{24}{7} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{-5}{6}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-120& = & -325x+1950 \\\Leftrightarrow & 78x \color{red}{-120} \color{blue}{+120} \color{blue}{+325x} & = & \color{red}{-325x} +1950 \color{blue}{+325x} \color{blue}{+120} \\\Leftrightarrow & 403x& = & 2070 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{2070}{403} \\\Leftrightarrow & x = \frac{2070}{403} & & \\ & V = \left\{ \frac{2070}{403} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & -546x+2730 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{+546x} & = & \color{red}{-546x} +2730 \color{blue}{+546x} \color{blue}{-150} \\\Leftrightarrow & 611x& = & 2580 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{2580}{611} \\\Leftrightarrow & x = \frac{2580}{611} & & \\ & V = \left\{ \frac{2580}{611} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 80x-120 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-80x} & = & \color{red}{80x} -120 \color{blue}{-80x} \color{blue}{+18} \\\Leftrightarrow & -65x& = & -102 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-102}{-65} \\\Leftrightarrow & x = \frac{102}{65} & & \\ & V = \left\{ \frac{102}{65} \right\} & \\\end{align}\)
12. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{-7}{4}x+2 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }+ \frac{ 56 }{ \color{blue}{308} })& = & (\frac{-539}{ \color{blue}{308} }x+\frac{616}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x+56& = & -539x+616 \\\Leftrightarrow & 44x \color{red}{+56} \color{blue}{-56} \color{blue}{+539x} & = & \color{red}{-539x} +616 \color{blue}{+539x} \color{blue}{-56} \\\Leftrightarrow & 583x& = & 560 \\\Leftrightarrow & \frac{583x}{ \color{red}{583} }& = & \frac{560}{583} \\\Leftrightarrow & x = \frac{560}{583} & & \\ & V = \left\{ \frac{560}{583} \right\} & \\\end{align}\)