Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{2}{13}=\frac{5}{6}x+5\)
2. \(\frac{x}{3}-\frac{5}{11}=\frac{1}{4}x-6\)
3. \(\frac{x}{3}-\frac{2}{7}=\frac{-3}{4}x-6\)
4. \(\frac{x}{6}+\frac{3}{16}=\frac{-7}{5}x+5\)
5. \(\frac{x}{2}-\frac{2}{15}=\frac{7}{3}x+6\)
6. \(\frac{x}{5}+\frac{2}{11}=\frac{5}{6}x+5\)
7. \(\frac{x}{6}+\frac{5}{6}=\frac{-7}{5}x+1\)
8. \(\frac{x}{6}+\frac{2}{7}=\frac{2}{5}x-4\)
9. \(\frac{x}{2}+\frac{4}{15}=\frac{2}{3}x-4\)
10. \(\frac{x}{2}+\frac{2}{9}=\frac{-2}{3}x+6\)
11. \(\frac{x}{2}-\frac{5}{7}=\frac{-5}{3}x-1\)
12. \(\frac{x}{6}+\frac{5}{14}=\frac{3}{5}x-2\)

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

Verbetersleutel

1. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{5}{6}x+5 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 84 }{ \color{blue}{546} })& = & (\frac{455}{ \color{blue}{546} }x+\frac{2730}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+84& = & 455x+2730 \\\Leftrightarrow & 78x \color{red}{+84} \color{blue}{-84} \color{blue}{-455x} & = & \color{red}{455x} +2730 \color{blue}{-455x} \color{blue}{-84} \\\Leftrightarrow & -377x& = & 2646 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{2646}{-377} \\\Leftrightarrow & x = \frac{-2646}{377} & & \\ & V = \left\{ \frac{-2646}{377} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x-\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-60& = & 33x-792 \\\Leftrightarrow & 44x \color{red}{-60} \color{blue}{+60} \color{blue}{-33x} & = & \color{red}{33x} -792 \color{blue}{-33x} \color{blue}{+60} \\\Leftrightarrow & 11x& = & -732 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-732}{11} \\\Leftrightarrow & x = \frac{-732}{11} & & \\ & V = \left\{ \frac{-732}{11} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & -63x-504 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{+63x} & = & \color{red}{-63x} -504 \color{blue}{+63x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & -480 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-480}{91} \\\Leftrightarrow & x = \frac{-480}{91} & & \\ & V = \left\{ \frac{-480}{91} \right\} & \\\end{align}\)
4. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x+\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -336x+1200 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+336x} & = & \color{red}{-336x} +1200 \color{blue}{+336x} \color{blue}{-45} \\\Leftrightarrow & 376x& = & 1155 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{1155}{376} \\\Leftrightarrow & x = \frac{1155}{376} & & \\ & V = \left\{ \frac{1155}{376} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{7}{3}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & 70x+180 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{-70x} & = & \color{red}{70x} +180 \color{blue}{-70x} \color{blue}{+4} \\\Leftrightarrow & -55x& = & 184 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{184}{-55} \\\Leftrightarrow & x = \frac{-184}{55} & & \\ & V = \left\{ \frac{-184}{55} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{5}{6}x+5 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{275}{ \color{blue}{330} }x+\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+60& = & 275x+1650 \\\Leftrightarrow & 66x \color{red}{+60} \color{blue}{-60} \color{blue}{-275x} & = & \color{red}{275x} +1650 \color{blue}{-275x} \color{blue}{-60} \\\Leftrightarrow & -209x& = & 1590 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{1590}{-209} \\\Leftrightarrow & x = \frac{-1590}{209} & & \\ & V = \left\{ \frac{-1590}{209} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -42x+30 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+42x} & = & \color{red}{-42x} +30 \color{blue}{+42x} \color{blue}{-25} \\\Leftrightarrow & 47x& = & 5 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{5}{47} \\\Leftrightarrow & x = \frac{5}{47} & & \\ & V = \left\{ \frac{5}{47} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{2}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 84x-840 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-84x} & = & \color{red}{84x} -840 \color{blue}{-84x} \color{blue}{-60} \\\Leftrightarrow & -49x& = & -900 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-900}{-49} \\\Leftrightarrow & x = \frac{900}{49} & & \\ & V = \left\{ \frac{900}{49} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{2}{3}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{20}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 20x-120 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-20x} & = & \color{red}{20x} -120 \color{blue}{-20x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & -128 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-128}{-5} \\\Leftrightarrow & x = \frac{128}{5} & & \\ & V = \left\{ \frac{128}{5} \right\} & \\\end{align}\)
10. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x+\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & -12x+108 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{+12x} & = & \color{red}{-12x} +108 \color{blue}{+12x} \color{blue}{-4} \\\Leftrightarrow & 21x& = & 104 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{104}{21} \\\Leftrightarrow & x = \frac{104}{21} & & \\ & V = \left\{ \frac{104}{21} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & -70x-42 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{+70x} & = & \color{red}{-70x} -42 \color{blue}{+70x} \color{blue}{+30} \\\Leftrightarrow & 91x& = & -12 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-12}{91} \\\Leftrightarrow & x = \frac{-12}{91} & & \\ & V = \left\{ \frac{-12}{91} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{3}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & 126x-420 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{-126x} & = & \color{red}{126x} -420 \color{blue}{-126x} \color{blue}{-75} \\\Leftrightarrow & -91x& = & -495 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-495}{-91} \\\Leftrightarrow & x = \frac{495}{91} & & \\ & V = \left\{ \frac{495}{91} \right\} & \\\end{align}\)