Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-2}{5}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -156x-1560 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+156x} & = & \color{red}{-156x} -1560 \color{blue}{+156x} \color{blue}{-90} \\\Leftrightarrow & 221x& = & -1650 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-1650}{221} \\\Leftrightarrow & x = \frac{-1650}{221} & & \\ & V = \left\{ \frac{-1650}{221} \right\} & \\\end{align}\)
2. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{4}x-8 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }+ \frac{ 80 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x-\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x+80& = & 65x-2080 \\\Leftrightarrow & 52x \color{red}{+80} \color{blue}{-80} \color{blue}{-65x} & = & \color{red}{65x} -2080 \color{blue}{-65x} \color{blue}{-80} \\\Leftrightarrow & -13x& = & -2160 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-2160}{-13} \\\Leftrightarrow & x = \frac{2160}{13} & & \\ & V = \left\{ \frac{2160}{13} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 30 }{ \color{blue}{105} })& = & (\frac{126}{ \color{blue}{105} }x-\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-30& = & 126x-525 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{-126x} & = & \color{red}{126x} -525 \color{blue}{-126x} \color{blue}{+30} \\\Leftrightarrow & -91x& = & -495 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-495}{-91} \\\Leftrightarrow & x = \frac{495}{91} & & \\ & V = \left\{ \frac{495}{91} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 6x-90 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-6x} & = & \color{red}{6x} -90 \color{blue}{-6x} \color{blue}{+8} \\\Leftrightarrow & -x& = & -82 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-82}{-1} \\\Leftrightarrow & x = 82 & & \\ & V = \left\{ 82 \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x-\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x+45& = & 288x-960 \\\Leftrightarrow & 80x \color{red}{+45} \color{blue}{-45} \color{blue}{-288x} & = & \color{red}{288x} -960 \color{blue}{-288x} \color{blue}{-45} \\\Leftrightarrow & -208x& = & -1005 \\\Leftrightarrow & \frac{-208x}{ \color{red}{-208} }& = & \frac{-1005}{-208} \\\Leftrightarrow & x = \frac{1005}{208} & & \\ & V = \left\{ \frac{1005}{208} \right\} & \\\end{align}\)
6. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x-\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-40& = & -52x-260 \\\Leftrightarrow & 65x \color{red}{-40} \color{blue}{+40} \color{blue}{+52x} & = & \color{red}{-52x} -260 \color{blue}{+52x} \color{blue}{+40} \\\Leftrightarrow & 117x& = & -220 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-220}{117} \\\Leftrightarrow & x = \frac{-220}{117} & & \\ & V = \left\{ \frac{-220}{117} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -12x+120 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+12x} & = & \color{red}{-12x} +120 \color{blue}{+12x} \color{blue}{+4} \\\Leftrightarrow & 17x& = & 124 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{124}{17} \\\Leftrightarrow & x = \frac{124}{17} & & \\ & V = \left\{ \frac{124}{17} \right\} & \\\end{align}\)
8. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{3}{4}x+2 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }+ \frac{ 60 }{ \color{blue}{220} })& = & (\frac{165}{ \color{blue}{220} }x+\frac{440}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x+60& = & 165x+440 \\\Leftrightarrow & 44x \color{red}{+60} \color{blue}{-60} \color{blue}{-165x} & = & \color{red}{165x} +440 \color{blue}{-165x} \color{blue}{-60} \\\Leftrightarrow & -121x& = & 380 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{380}{-121} \\\Leftrightarrow & x = \frac{-380}{121} & & \\ & V = \left\{ \frac{-380}{121} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x+\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+24& = & -104x+1248 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+104x} & = & \color{red}{-104x} +1248 \color{blue}{+104x} \color{blue}{-24} \\\Leftrightarrow & 143x& = & 1224 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{1224}{143} \\\Leftrightarrow & x = \frac{1224}{143} & & \\ & V = \left\{ \frac{1224}{143} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & -546x+2730 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{+546x} & = & \color{red}{-546x} +2730 \color{blue}{+546x} \color{blue}{+120} \\\Leftrightarrow & 611x& = & 2850 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{2850}{611} \\\Leftrightarrow & x = \frac{2850}{611} & & \\ & V = \left\{ \frac{2850}{611} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & 44x-924 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-44x} & = & \color{red}{44x} -924 \color{blue}{-44x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & -948 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-948}{-11} \\\Leftrightarrow & x = \frac{948}{11} & & \\ & V = \left\{ \frac{948}{11} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-5}{6}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & -175x-210 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{+175x} & = & \color{red}{-175x} -210 \color{blue}{+175x} \color{blue}{-90} \\\Leftrightarrow & 217x& = & -300 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-300}{217} \\\Leftrightarrow & x = \frac{-300}{217} & & \\ & V = \left\{ \frac{-300}{217} \right\} & \\\end{align}\)