Vgln. eerste graad (reeks 4)

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

1. \(\frac{x}{5}+\frac{4}{17}=\frac{-3}{7}x+8\)
2. \(\frac{x}{7}+\frac{2}{15}=\frac{-8}{3}x+8\)
3. \(\frac{x}{2}+\frac{5}{16}=\frac{8}{3}x-1\)
4. \(\frac{x}{5}+\frac{4}{7}=\frac{1}{4}x-7\)
5. \(\frac{x}{2}+\frac{5}{12}=\frac{1}{3}x+1\)
6. \(\frac{x}{4}+\frac{3}{10}=\frac{8}{7}x+7\)
7. \(\frac{x}{5}+\frac{2}{9}=\frac{3}{4}x+3\)
8. \(\frac{x}{6}-\frac{2}{11}=\frac{2}{5}x+8\)
9. \(\frac{x}{2}+\frac{4}{17}=\frac{-8}{3}x-7\)
10. \(\frac{x}{2}-\frac{2}{9}=\frac{2}{5}x+3\)
11. \(\frac{x}{6}+\frac{4}{15}=\frac{-6}{7}x+5\)
12. \(\frac{x}{3}+\frac{4}{17}=\frac{1}{2}x-7\)

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

Verbetersleutel

1. \(\text{595 is het kleinste gemene veelvoud van 5, 17 en 7} \\ \begin{align} & \color{blue}{595.} (\frac{x}{5}+\frac{4}{17})& = & (\frac{-3}{7}x+8) \color{blue}{.595} \\\Leftrightarrow & 119x+140& = & -255x+4760 \\\Leftrightarrow & 119x \color{red}{+140} \color{blue}{-140} \color{blue}{+255x} & = & \color{red}{-255x} +4760 \color{blue}{+255x} \color{blue}{-140} \\\Leftrightarrow & 374x& = & 4620 \\\Leftrightarrow & \frac{374x}{ \color{red}{374} }& = & \frac{4620}{374} \\\Leftrightarrow & x = \frac{210}{17} & & \\ & V = \left\{ \frac{210}{17} \right\} & \\\end{align}\)
2. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \color{blue}{105.} (\frac{x}{7}+\frac{2}{15})& = & (\frac{-8}{3}x+8) \color{blue}{.105} \\\Leftrightarrow & 15x+14& = & -280x+840 \\\Leftrightarrow & 15x \color{red}{+14} \color{blue}{-14} \color{blue}{+280x} & = & \color{red}{-280x} +840 \color{blue}{+280x} \color{blue}{-14} \\\Leftrightarrow & 295x& = & 826 \\\Leftrightarrow & \frac{295x}{ \color{red}{295} }& = & \frac{826}{295} \\\Leftrightarrow & x = \frac{14}{5} & & \\ & V = \left\{ \frac{14}{5} \right\} & \\\end{align}\)
3. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \color{blue}{48.} (\frac{x}{2}+\frac{5}{16})& = & (\frac{8}{3}x-1) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 128x-48 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-128x} & = & \color{red}{128x} -48 \color{blue}{-128x} \color{blue}{-15} \\\Leftrightarrow & -104x& = & -63 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{-63}{-104} \\\Leftrightarrow & x = \frac{63}{104} & & \\ & V = \left\{ \frac{63}{104} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \color{blue}{140.} (\frac{x}{5}+\frac{4}{7})& = & (\frac{1}{4}x-7) \color{blue}{.140} \\\Leftrightarrow & 28x+80& = & 35x-980 \\\Leftrightarrow & 28x \color{red}{+80} \color{blue}{-80} \color{blue}{-35x} & = & \color{red}{35x} -980 \color{blue}{-35x} \color{blue}{-80} \\\Leftrightarrow & -7x& = & -1060 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-1060}{-7} \\\Leftrightarrow & x = \frac{1060}{7} & & \\ & V = \left\{ \frac{1060}{7} \right\} & \\\end{align}\)
5. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \color{blue}{12.} (\frac{x}{2}+\frac{5}{12})& = & (\frac{1}{3}x+1) \color{blue}{.12} \\\Leftrightarrow & 6x+5& = & 4x+12 \\\Leftrightarrow & 6x \color{red}{+5} \color{blue}{-5} \color{blue}{-4x} & = & \color{red}{4x} +12 \color{blue}{-4x} \color{blue}{-5} \\\Leftrightarrow & 2x& = & 7 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{7}{2} \\\Leftrightarrow & x = \frac{7}{2} & & \\ & V = \left\{ \frac{7}{2} \right\} & \\\end{align}\)
6. \(\text{140 is het kleinste gemene veelvoud van 4, 10 en 7} \\ \begin{align} & \color{blue}{140.} (\frac{x}{4}+\frac{3}{10})& = & (\frac{8}{7}x+7) \color{blue}{.140} \\\Leftrightarrow & 35x+42& = & 160x+980 \\\Leftrightarrow & 35x \color{red}{+42} \color{blue}{-42} \color{blue}{-160x} & = & \color{red}{160x} +980 \color{blue}{-160x} \color{blue}{-42} \\\Leftrightarrow & -125x& = & 938 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{938}{-125} \\\Leftrightarrow & x = \frac{-938}{125} & & \\ & V = \left\{ \frac{-938}{125} \right\} & \\\end{align}\)
7. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \color{blue}{180.} (\frac{x}{5}+\frac{2}{9})& = & (\frac{3}{4}x+3) \color{blue}{.180} \\\Leftrightarrow & 36x+40& = & 135x+540 \\\Leftrightarrow & 36x \color{red}{+40} \color{blue}{-40} \color{blue}{-135x} & = & \color{red}{135x} +540 \color{blue}{-135x} \color{blue}{-40} \\\Leftrightarrow & -99x& = & 500 \\\Leftrightarrow & \frac{-99x}{ \color{red}{-99} }& = & \frac{500}{-99} \\\Leftrightarrow & x = \frac{-500}{99} & & \\ & V = \left\{ \frac{-500}{99} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \color{blue}{330.} (\frac{x}{6}-\frac{2}{11})& = & (\frac{2}{5}x+8) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & 132x+2640 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-132x} & = & \color{red}{132x} +2640 \color{blue}{-132x} \color{blue}{+60} \\\Leftrightarrow & -77x& = & 2700 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{2700}{-77} \\\Leftrightarrow & x = \frac{-2700}{77} & & \\ & V = \left\{ \frac{-2700}{77} \right\} & \\\end{align}\)
9. \(\text{102 is het kleinste gemene veelvoud van 2, 17 en 3} \\ \begin{align} & \color{blue}{102.} (\frac{x}{2}+\frac{4}{17})& = & (\frac{-8}{3}x-7) \color{blue}{.102} \\\Leftrightarrow & 51x+24& = & -272x-714 \\\Leftrightarrow & 51x \color{red}{+24} \color{blue}{-24} \color{blue}{+272x} & = & \color{red}{-272x} -714 \color{blue}{+272x} \color{blue}{-24} \\\Leftrightarrow & 323x& = & -738 \\\Leftrightarrow & \frac{323x}{ \color{red}{323} }& = & \frac{-738}{323} \\\Leftrightarrow & x = \frac{-738}{323} & & \\ & V = \left\{ \frac{-738}{323} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \color{blue}{90.} (\frac{x}{2}-\frac{2}{9})& = & (\frac{2}{5}x+3) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & 36x+270 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{-36x} & = & \color{red}{36x} +270 \color{blue}{-36x} \color{blue}{+20} \\\Leftrightarrow & 9x& = & 290 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{290}{9} \\\Leftrightarrow & x = \frac{290}{9} & & \\ & V = \left\{ \frac{290}{9} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 15 en 7} \\ \begin{align} & \color{blue}{210.} (\frac{x}{6}+\frac{4}{15})& = & (\frac{-6}{7}x+5) \color{blue}{.210} \\\Leftrightarrow & 35x+56& = & -180x+1050 \\\Leftrightarrow & 35x \color{red}{+56} \color{blue}{-56} \color{blue}{+180x} & = & \color{red}{-180x} +1050 \color{blue}{+180x} \color{blue}{-56} \\\Leftrightarrow & 215x& = & 994 \\\Leftrightarrow & \frac{215x}{ \color{red}{215} }& = & \frac{994}{215} \\\Leftrightarrow & x = \frac{994}{215} & & \\ & V = \left\{ \frac{994}{215} \right\} & \\\end{align}\)
12. \(\text{102 is het kleinste gemene veelvoud van 3, 17 en 2} \\ \begin{align} & \color{blue}{102.} (\frac{x}{3}+\frac{4}{17})& = & (\frac{1}{2}x-7) \color{blue}{.102} \\\Leftrightarrow & 34x+24& = & 51x-714 \\\Leftrightarrow & 34x \color{red}{+24} \color{blue}{-24} \color{blue}{-51x} & = & \color{red}{51x} -714 \color{blue}{-51x} \color{blue}{-24} \\\Leftrightarrow & -17x& = & -738 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{-738}{-17} \\\Leftrightarrow & x = \frac{738}{17} & & \\ & V = \left\{ \frac{738}{17} \right\} & \\\end{align}\)