Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{1}{6}x+4 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{20}{ \color{blue}{120} }x+\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x+75& = & 20x+480 \\\Leftrightarrow & 24x \color{red}{+75} \color{blue}{-75} \color{blue}{-20x} & = & \color{red}{20x} +480 \color{blue}{-20x} \color{blue}{-75} \\\Leftrightarrow & 4x& = & 405 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{405}{4} \\\Leftrightarrow & x = \frac{405}{4} & & \\ & V = \left\{ \frac{405}{4} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{7}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+150& = & 35x+210 \\\Leftrightarrow & 42x \color{red}{+150} \color{blue}{-150} \color{blue}{-35x} & = & \color{red}{35x} +210 \color{blue}{-35x} \color{blue}{-150} \\\Leftrightarrow & 7x& = & 60 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{60}{7} \\\Leftrightarrow & x = \frac{60}{7} & & \\ & V = \left\{ \frac{60}{7} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 28x-980 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-28x} & = & \color{red}{28x} -980 \color{blue}{-28x} \color{blue}{-40} \\\Leftrightarrow & 7x& = & -1020 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1020}{7} \\\Leftrightarrow & x = \frac{-1020}{7} & & \\ & V = \left\{ \frac{-1020}{7} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & -84x-210 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{+84x} & = & \color{red}{-84x} -210 \color{blue}{+84x} \color{blue}{-75} \\\Leftrightarrow & 119x& = & -285 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-285}{119} \\\Leftrightarrow & x = \frac{-285}{119} & & \\ & V = \left\{ \frac{-285}{119} \right\} & \\\end{align}\)
5. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{2}{3}x+4 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{16}{ \color{blue}{24} }x+\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+15& = & 16x+96 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{-16x} & = & \color{red}{16x} +96 \color{blue}{-16x} \color{blue}{-15} \\\Leftrightarrow & -4x& = & 81 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{81}{-4} \\\Leftrightarrow & x = \frac{-81}{4} & & \\ & V = \left\{ \frac{-81}{4} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{1}{4}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+18& = & 15x-300 \\\Leftrightarrow & 20x \color{red}{+18} \color{blue}{-18} \color{blue}{-15x} & = & \color{red}{15x} -300 \color{blue}{-15x} \color{blue}{-18} \\\Leftrightarrow & 5x& = & -318 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{-318}{5} \\\Leftrightarrow & x = \frac{-318}{5} & & \\ & V = \left\{ \frac{-318}{5} \right\} & \\\end{align}\)
7. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{-7}{4}x-7 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }+ \frac{ 84 }{ \color{blue}{364} })& = & (\frac{-637}{ \color{blue}{364} }x-\frac{2548}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x+84& = & -637x-2548 \\\Leftrightarrow & 52x \color{red}{+84} \color{blue}{-84} \color{blue}{+637x} & = & \color{red}{-637x} -2548 \color{blue}{+637x} \color{blue}{-84} \\\Leftrightarrow & 689x& = & -2632 \\\Leftrightarrow & \frac{689x}{ \color{red}{689} }& = & \frac{-2632}{689} \\\Leftrightarrow & x = \frac{-2632}{689} & & \\ & V = \left\{ \frac{-2632}{689} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -42x+30 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+42x} & = & \color{red}{-42x} +30 \color{blue}{+42x} \color{blue}{-25} \\\Leftrightarrow & 47x& = & 5 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{5}{47} \\\Leftrightarrow & x = \frac{5}{47} & & \\ & V = \left\{ \frac{5}{47} \right\} & \\\end{align}\)
9. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{4}{5}x+7 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{64}{ \color{blue}{80} }x+\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+25& = & 64x+560 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{-64x} & = & \color{red}{64x} +560 \color{blue}{-64x} \color{blue}{-25} \\\Leftrightarrow & -44x& = & 535 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{535}{-44} \\\Leftrightarrow & x = \frac{-535}{44} & & \\ & V = \left\{ \frac{-535}{44} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -56x+140 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+56x} & = & \color{red}{-56x} +140 \color{blue}{+56x} \color{blue}{+20} \\\Leftrightarrow & 91x& = & 160 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{160}{91} \\\Leftrightarrow & x = \frac{160}{91} & & \\ & V = \left\{ \frac{160}{91} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+30& = & 35x+210 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-35x} & = & \color{red}{35x} +210 \color{blue}{-35x} \color{blue}{-30} \\\Leftrightarrow & -14x& = & 180 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{180}{-14} \\\Leftrightarrow & x = \frac{-90}{7} & & \\ & V = \left\{ \frac{-90}{7} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{5}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 105x-210 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-105x} & = & \color{red}{105x} -210 \color{blue}{-105x} \color{blue}{-12} \\\Leftrightarrow & -91x& = & -222 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-222}{-91} \\\Leftrightarrow & x = \frac{222}{91} & & \\ & V = \left\{ \frac{222}{91} \right\} & \\\end{align}\)