Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-5}{6}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-12& = & -35x+42 \\\Leftrightarrow & 6x \color{red}{-12} \color{blue}{+12} \color{blue}{+35x} & = & \color{red}{-35x} +42 \color{blue}{+35x} \color{blue}{+12} \\\Leftrightarrow & 41x& = & 54 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{54}{41} \\\Leftrightarrow & x = \frac{54}{41} & & \\ & V = \left\{ \frac{54}{41} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 15x+210 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} +210 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -9x& = & 214 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{214}{-9} \\\Leftrightarrow & x = \frac{-214}{9} & & \\ & V = \left\{ \frac{-214}{9} \right\} & \\\end{align}\)
3. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{-3}{4}x-4 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }+ \frac{ 140 }{ \color{blue}{252} })& = & (\frac{-189}{ \color{blue}{252} }x-\frac{1008}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x+140& = & -189x-1008 \\\Leftrightarrow & 36x \color{red}{+140} \color{blue}{-140} \color{blue}{+189x} & = & \color{red}{-189x} -1008 \color{blue}{+189x} \color{blue}{-140} \\\Leftrightarrow & 225x& = & -1148 \\\Leftrightarrow & \frac{225x}{ \color{red}{225} }& = & \frac{-1148}{225} \\\Leftrightarrow & x = \frac{-1148}{225} & & \\ & V = \left\{ \frac{-1148}{225} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{5}{4}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-16& = & 75x+240 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{-75x} & = & \color{red}{75x} +240 \color{blue}{-75x} \color{blue}{+16} \\\Leftrightarrow & -63x& = & 256 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{256}{-63} \\\Leftrightarrow & x = \frac{-256}{63} & & \\ & V = \left\{ \frac{-256}{63} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 88x+66 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-88x} & = & \color{red}{88x} +66 \color{blue}{-88x} \color{blue}{+12} \\\Leftrightarrow & -55x& = & 78 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{78}{-55} \\\Leftrightarrow & x = \frac{-78}{55} & & \\ & V = \left\{ \frac{-78}{55} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 66x-330 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-66x} & = & \color{red}{66x} -330 \color{blue}{-66x} \color{blue}{-120} \\\Leftrightarrow & -11x& = & -450 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-450}{-11} \\\Leftrightarrow & x = \frac{450}{11} & & \\ & V = \left\{ \frac{450}{11} \right\} & \\\end{align}\)
7. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 12 }{ \color{blue}{21} })& = & (\frac{-14}{ \color{blue}{21} }x+\frac{42}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-12& = & -14x+42 \\\Leftrightarrow & 3x \color{red}{-12} \color{blue}{+12} \color{blue}{+14x} & = & \color{red}{-14x} +42 \color{blue}{+14x} \color{blue}{+12} \\\Leftrightarrow & 17x& = & 54 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{54}{17} \\\Leftrightarrow & x = \frac{54}{17} & & \\ & V = \left\{ \frac{54}{17} \right\} & \\\end{align}\)
8. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x-\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & -24x-36 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{+24x} & = & \color{red}{-24x} -36 \color{blue}{+24x} \color{blue}{+16} \\\Leftrightarrow & 33x& = & -20 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-20}{33} \\\Leftrightarrow & x = \frac{-20}{33} & & \\ & V = \left\{ \frac{-20}{33} \right\} & \\\end{align}\)
9. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 4 }{ \color{blue}{15} })& = & (\frac{-12}{ \color{blue}{15} }x+\frac{105}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+4& = & -12x+105 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+12x} & = & \color{red}{-12x} +105 \color{blue}{+12x} \color{blue}{-4} \\\Leftrightarrow & 17x& = & 101 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{101}{17} \\\Leftrightarrow & x = \frac{101}{17} & & \\ & V = \left\{ \frac{101}{17} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & -50x-60 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{+50x} & = & \color{red}{-50x} -60 \color{blue}{+50x} \color{blue}{+9} \\\Leftrightarrow & 65x& = & -51 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{-51}{65} \\\Leftrightarrow & x = \frac{-51}{65} & & \\ & V = \left\{ \frac{-51}{65} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 21x+168 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-21x} & = & \color{red}{21x} +168 \color{blue}{-21x} \color{blue}{+36} \\\Leftrightarrow & 7x& = & 204 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{204}{7} \\\Leftrightarrow & x = \frac{204}{7} & & \\ & V = \left\{ \frac{204}{7} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x-\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & -112x-840 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+112x} & = & \color{red}{-112x} -840 \color{blue}{+112x} \color{blue}{+40} \\\Leftrightarrow & 147x& = & -800 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{-800}{147} \\\Leftrightarrow & x = \frac{-800}{147} & & \\ & V = \left\{ \frac{-800}{147} \right\} & \\\end{align}\)