Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{182}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 91x+182 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-91x} & = & \color{red}{91x} +182 \color{blue}{-91x} \color{blue}{+42} \\\Leftrightarrow & -65x& = & 224 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{224}{-65} \\\Leftrightarrow & x = \frac{-224}{65} & & \\ & V = \left\{ \frac{-224}{65} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{3}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 18x+90 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-18x} & = & \color{red}{18x} +90 \color{blue}{-18x} \color{blue}{+4} \\\Leftrightarrow & -13x& = & 94 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{94}{-13} \\\Leftrightarrow & x = \frac{-94}{13} & & \\ & V = \left\{ \frac{-94}{13} \right\} & \\\end{align}\)
3. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{3}{5}x-4 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{198}{ \color{blue}{330} }x-\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 198x-1320 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-198x} & = & \color{red}{198x} -1320 \color{blue}{-198x} \color{blue}{-120} \\\Leftrightarrow & -143x& = & -1440 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-1440}{-143} \\\Leftrightarrow & x = \frac{1440}{143} & & \\ & V = \left\{ \frac{1440}{143} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -24x-240 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+24x} & = & \color{red}{-24x} -240 \color{blue}{+24x} \color{blue}{+8} \\\Leftrightarrow & 29x& = & -232 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-232}{29} \\\Leftrightarrow & x = -8 & & \\ & V = \left\{ -8 \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -28x+210 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+28x} & = & \color{red}{-28x} +210 \color{blue}{+28x} \color{blue}{+18} \\\Leftrightarrow & 49x& = & 228 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{228}{49} \\\Leftrightarrow & x = \frac{228}{49} & & \\ & V = \left\{ \frac{228}{49} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{4}{5}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 168x+1260 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-168x} & = & \color{red}{168x} +1260 \color{blue}{-168x} \color{blue}{+90} \\\Leftrightarrow & -133x& = & 1350 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{1350}{-133} \\\Leftrightarrow & x = \frac{-1350}{133} & & \\ & V = \left\{ \frac{-1350}{133} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 12 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{4}{5}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{48}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+25& = & 48x-480 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{-48x} & = & \color{red}{48x} -480 \color{blue}{-48x} \color{blue}{-25} \\\Leftrightarrow & -33x& = & -505 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-505}{-33} \\\Leftrightarrow & x = \frac{505}{33} & & \\ & V = \left\{ \frac{505}{33} \right\} & \\\end{align}\)
8. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 45 }{ \color{blue}{165} })& = & (\frac{-110}{ \color{blue}{165} }x+\frac{660}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+45& = & -110x+660 \\\Leftrightarrow & 33x \color{red}{+45} \color{blue}{-45} \color{blue}{+110x} & = & \color{red}{-110x} +660 \color{blue}{+110x} \color{blue}{-45} \\\Leftrightarrow & 143x& = & 615 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{615}{143} \\\Leftrightarrow & x = \frac{615}{143} & & \\ & V = \left\{ \frac{615}{143} \right\} & \\\end{align}\)
9. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{5}{6}x-5 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 210 }{ \color{blue}{462} })& = & (\frac{385}{ \color{blue}{462} }x-\frac{2310}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-210& = & 385x-2310 \\\Leftrightarrow & 66x \color{red}{-210} \color{blue}{+210} \color{blue}{-385x} & = & \color{red}{385x} -2310 \color{blue}{-385x} \color{blue}{+210} \\\Leftrightarrow & -319x& = & -2100 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{-2100}{-319} \\\Leftrightarrow & x = \frac{2100}{319} & & \\ & V = \left\{ \frac{2100}{319} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & -312x+3120 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{+312x} & = & \color{red}{-312x} +3120 \color{blue}{+312x} \color{blue}{+60} \\\Leftrightarrow & 377x& = & 3180 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{3180}{377} \\\Leftrightarrow & x = \frac{3180}{377} & & \\ & V = \left\{ \frac{3180}{377} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{7}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & 294x-210 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{-294x} & = & \color{red}{294x} -210 \color{blue}{-294x} \color{blue}{-45} \\\Leftrightarrow & -259x& = & -255 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{-255}{-259} \\\Leftrightarrow & x = \frac{255}{259} & & \\ & V = \left\{ \frac{255}{259} \right\} & \\\end{align}\)
12. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & -104x+624 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+104x} & = & \color{red}{-104x} +624 \color{blue}{+104x} \color{blue}{+24} \\\Leftrightarrow & 143x& = & 648 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{648}{143} \\\Leftrightarrow & x = \frac{648}{143} & & \\ & V = \left\{ \frac{648}{143} \right\} & \\\end{align}\)