Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & -126x+360 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{+126x} & = & \color{red}{-126x} +360 \color{blue}{+126x} \color{blue}{+50} \\\Leftrightarrow & 141x& = & 410 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{410}{141} \\\Leftrightarrow & x = \frac{410}{141} & & \\ & V = \left\{ \frac{410}{141} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-5}{6}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+24& = & -35x+42 \\\Leftrightarrow & 6x \color{red}{+24} \color{blue}{-24} \color{blue}{+35x} & = & \color{red}{-35x} +42 \color{blue}{+35x} \color{blue}{-24} \\\Leftrightarrow & 41x& = & 18 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{18}{41} \\\Leftrightarrow & x = \frac{18}{41} & & \\ & V = \left\{ \frac{18}{41} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 4, 12 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+25& = & 12x-300 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{-12x} & = & \color{red}{12x} -300 \color{blue}{-12x} \color{blue}{-25} \\\Leftrightarrow & 3x& = & -325 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-325}{3} \\\Leftrightarrow & x = \frac{-325}{3} & & \\ & V = \left\{ \frac{-325}{3} \right\} & \\\end{align}\)
4. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x-\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x+75& = & -192x-1440 \\\Leftrightarrow & 80x \color{red}{+75} \color{blue}{-75} \color{blue}{+192x} & = & \color{red}{-192x} -1440 \color{blue}{+192x} \color{blue}{-75} \\\Leftrightarrow & 272x& = & -1515 \\\Leftrightarrow & \frac{272x}{ \color{red}{272} }& = & \frac{-1515}{272} \\\Leftrightarrow & x = \frac{-1515}{272} & & \\ & V = \left\{ \frac{-1515}{272} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{5}{3}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{260}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+24& = & 260x-780 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-260x} & = & \color{red}{260x} -780 \color{blue}{-260x} \color{blue}{-24} \\\Leftrightarrow & -221x& = & -804 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{-804}{-221} \\\Leftrightarrow & x = \frac{804}{221} & & \\ & V = \left\{ \frac{804}{221} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 5x+240 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-5x} & = & \color{red}{5x} +240 \color{blue}{-5x} \color{blue}{-4} \\\Leftrightarrow & x& = & 236 \\ & V = \left\{ 236 \right\} & \\\end{align}\)
7. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 21 }{ \color{blue}{112} })& = & (\frac{280}{ \color{blue}{112} }x-\frac{784}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+21& = & 280x-784 \\\Leftrightarrow & 16x \color{red}{+21} \color{blue}{-21} \color{blue}{-280x} & = & \color{red}{280x} -784 \color{blue}{-280x} \color{blue}{-21} \\\Leftrightarrow & -264x& = & -805 \\\Leftrightarrow & \frac{-264x}{ \color{red}{-264} }& = & \frac{-805}{-264} \\\Leftrightarrow & x = \frac{805}{264} & & \\ & V = \left\{ \frac{805}{264} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x-\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -264x-2310 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+264x} & = & \color{red}{-264x} -2310 \color{blue}{+264x} \color{blue}{+150} \\\Leftrightarrow & 319x& = & -2160 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-2160}{319} \\\Leftrightarrow & x = \frac{-2160}{319} & & \\ & V = \left\{ \frac{-2160}{319} \right\} & \\\end{align}\)
9. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{5}{6}x+7 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 84 }{ \color{blue}{462} })& = & (\frac{385}{ \color{blue}{462} }x+\frac{3234}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-84& = & 385x+3234 \\\Leftrightarrow & 66x \color{red}{-84} \color{blue}{+84} \color{blue}{-385x} & = & \color{red}{385x} +3234 \color{blue}{-385x} \color{blue}{+84} \\\Leftrightarrow & -319x& = & 3318 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{3318}{-319} \\\Leftrightarrow & x = \frac{-3318}{319} & & \\ & V = \left\{ \frac{-3318}{319} \right\} & \\\end{align}\)
10. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{1274}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+42& = & 91x+1274 \\\Leftrightarrow & 26x \color{red}{+42} \color{blue}{-42} \color{blue}{-91x} & = & \color{red}{91x} +1274 \color{blue}{-91x} \color{blue}{-42} \\\Leftrightarrow & -65x& = & 1232 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{1232}{-65} \\\Leftrightarrow & x = \frac{-1232}{65} & & \\ & V = \left\{ \frac{-1232}{65} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & -42x-210 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+42x} & = & \color{red}{-42x} -210 \color{blue}{+42x} \color{blue}{+25} \\\Leftrightarrow & 47x& = & -185 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-185}{47} \\\Leftrightarrow & x = \frac{-185}{47} & & \\ & V = \left\{ \frac{-185}{47} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 14 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x+\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+14& = & 35x+735 \\\Leftrightarrow & 15x \color{red}{+14} \color{blue}{-14} \color{blue}{-35x} & = & \color{red}{35x} +735 \color{blue}{-35x} \color{blue}{-14} \\\Leftrightarrow & -20x& = & 721 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{721}{-20} \\\Leftrightarrow & x = \frac{-721}{20} & & \\ & V = \left\{ \frac{-721}{20} \right\} & \\\end{align}\)