Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{11}& = & \frac{7}{5}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{462}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-90& = & 462x-1980 \\\Leftrightarrow & 55x \color{red}{-90} \color{blue}{+90} \color{blue}{-462x} & = & \color{red}{462x} -1980 \color{blue}{-462x} \color{blue}{+90} \\\Leftrightarrow & -407x& = & -1890 \\\Leftrightarrow & \frac{-407x}{ \color{red}{-407} }& = & \frac{-1890}{-407} \\\Leftrightarrow & x = \frac{1890}{407} & & \\ & V = \left\{ \frac{1890}{407} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-208}{ \color{blue}{78} }x-\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & -208x-624 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{+208x} & = & \color{red}{-208x} -624 \color{blue}{+208x} \color{blue}{+18} \\\Leftrightarrow & 247x& = & -606 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-606}{247} \\\Leftrightarrow & x = \frac{-606}{247} & & \\ & V = \left\{ \frac{-606}{247} \right\} & \\\end{align}\)
3. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{-84}{ \color{blue}{112} }x-\frac{560}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & -84x-560 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{+84x} & = & \color{red}{-84x} -560 \color{blue}{+84x} \color{blue}{+35} \\\Leftrightarrow & 100x& = & -525 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{-525}{100} \\\Leftrightarrow & x = \frac{-21}{4} & & \\ & V = \left\{ \frac{-21}{4} \right\} & \\\end{align}\)
4. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{117}{ \color{blue}{195} }x+\frac{1365}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+45& = & 117x+1365 \\\Leftrightarrow & 65x \color{red}{+45} \color{blue}{-45} \color{blue}{-117x} & = & \color{red}{117x} +1365 \color{blue}{-117x} \color{blue}{-45} \\\Leftrightarrow & -52x& = & 1320 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{1320}{-52} \\\Leftrightarrow & x = \frac{-330}{13} & & \\ & V = \left\{ \frac{-330}{13} \right\} & \\\end{align}\)
5. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x+\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+60& = & -196x+1120 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+196x} & = & \color{red}{-196x} +1120 \color{blue}{+196x} \color{blue}{-60} \\\Leftrightarrow & 231x& = & 1060 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{1060}{231} \\\Leftrightarrow & x = \frac{1060}{231} & & \\ & V = \left\{ \frac{1060}{231} \right\} & \\\end{align}\)
6. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 84 }{ \color{blue}{364} })& = & (\frac{-273}{ \color{blue}{364} }x+\frac{2912}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-84& = & -273x+2912 \\\Leftrightarrow & 52x \color{red}{-84} \color{blue}{+84} \color{blue}{+273x} & = & \color{red}{-273x} +2912 \color{blue}{+273x} \color{blue}{+84} \\\Leftrightarrow & 325x& = & 2996 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{2996}{325} \\\Leftrightarrow & x = \frac{2996}{325} & & \\ & V = \left\{ \frac{2996}{325} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & -100x-240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+100x} & = & \color{red}{-100x} -240 \color{blue}{+100x} \color{blue}{+8} \\\Leftrightarrow & 115x& = & -232 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{-232}{115} \\\Leftrightarrow & x = \frac{-232}{115} & & \\ & V = \left\{ \frac{-232}{115} \right\} & \\\end{align}\)
8. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{4}{ \color{blue}{20} }x+\frac{140}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & 4x+140 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{-4x} & = & \color{red}{4x} +140 \color{blue}{-4x} \color{blue}{-6} \\\Leftrightarrow & x& = & 134 \\ & V = \left\{ 134 \right\} & \\\end{align}\)
9. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 28 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x-\frac{882}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-28& = & 63x-882 \\\Leftrightarrow & 18x \color{red}{-28} \color{blue}{+28} \color{blue}{-63x} & = & \color{red}{63x} -882 \color{blue}{-63x} \color{blue}{+28} \\\Leftrightarrow & -45x& = & -854 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-854}{-45} \\\Leftrightarrow & x = \frac{854}{45} & & \\ & V = \left\{ \frac{854}{45} \right\} & \\\end{align}\)
10. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-8}{ \color{blue}{12} }x+\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-5& = & -8x+96 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{+8x} & = & \color{red}{-8x} +96 \color{blue}{+8x} \color{blue}{+5} \\\Leftrightarrow & 11x& = & 101 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{101}{11} \\\Leftrightarrow & x = \frac{101}{11} & & \\ & V = \left\{ \frac{101}{11} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-105}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+8& = & -105x+240 \\\Leftrightarrow & 20x \color{red}{+8} \color{blue}{-8} \color{blue}{+105x} & = & \color{red}{-105x} +240 \color{blue}{+105x} \color{blue}{-8} \\\Leftrightarrow & 125x& = & 232 \\\Leftrightarrow & \frac{125x}{ \color{red}{125} }& = & \frac{232}{125} \\\Leftrightarrow & x = \frac{232}{125} & & \\ & V = \left\{ \frac{232}{125} \right\} & \\\end{align}\)
12. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x+\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & -60x+144 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+60x} & = & \color{red}{-60x} +144 \color{blue}{+60x} \color{blue}{+8} \\\Leftrightarrow & 69x& = & 152 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{152}{69} \\\Leftrightarrow & x = \frac{152}{69} & & \\ & V = \left\{ \frac{152}{69} \right\} & \\\end{align}\)