Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 210 }{ \color{blue}{462} })& = & (\frac{-385}{ \color{blue}{462} }x-\frac{3234}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-210& = & -385x-3234 \\\Leftrightarrow & 66x \color{red}{-210} \color{blue}{+210} \color{blue}{+385x} & = & \color{red}{-385x} -3234 \color{blue}{+385x} \color{blue}{+210} \\\Leftrightarrow & 451x& = & -3024 \\\Leftrightarrow & \frac{451x}{ \color{red}{451} }& = & \frac{-3024}{451} \\\Leftrightarrow & x = \frac{-3024}{451} & & \\ & V = \left\{ \frac{-3024}{451} \right\} & \\\end{align}\)
2. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 70 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x-\frac{252}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+70& = & 63x-252 \\\Leftrightarrow & 18x \color{red}{+70} \color{blue}{-70} \color{blue}{-63x} & = & \color{red}{63x} -252 \color{blue}{-63x} \color{blue}{-70} \\\Leftrightarrow & -45x& = & -322 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-322}{-45} \\\Leftrightarrow & x = \frac{322}{45} & & \\ & V = \left\{ \frac{322}{45} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & 14x+420 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-14x} & = & \color{red}{14x} +420 \color{blue}{-14x} \color{blue}{+40} \\\Leftrightarrow & 21x& = & 460 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{460}{21} \\\Leftrightarrow & x = \frac{460}{21} & & \\ & V = \left\{ \frac{460}{21} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & 55x+330 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{-55x} & = & \color{red}{55x} +330 \color{blue}{-55x} \color{blue}{-150} \\\Leftrightarrow & 11x& = & 180 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{180}{11} \\\Leftrightarrow & x = \frac{180}{11} & & \\ & V = \left\{ \frac{180}{11} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{13}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-150& = & -156x-2730 \\\Leftrightarrow & 65x \color{red}{-150} \color{blue}{+150} \color{blue}{+156x} & = & \color{red}{-156x} -2730 \color{blue}{+156x} \color{blue}{+150} \\\Leftrightarrow & 221x& = & -2580 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-2580}{221} \\\Leftrightarrow & x = \frac{-2580}{221} & & \\ & V = \left\{ \frac{-2580}{221} \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x-\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-60& = & -220x-924 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{+220x} & = & \color{red}{-220x} -924 \color{blue}{+220x} \color{blue}{+60} \\\Leftrightarrow & 253x& = & -864 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{-864}{253} \\\Leftrightarrow & x = \frac{-864}{253} & & \\ & V = \left\{ \frac{-864}{253} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 252x-420 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-252x} & = & \color{red}{252x} -420 \color{blue}{-252x} \color{blue}{+90} \\\Leftrightarrow & -217x& = & -330 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-330}{-217} \\\Leftrightarrow & x = \frac{330}{217} & & \\ & V = \left\{ \frac{330}{217} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -312x-1170 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+312x} & = & \color{red}{-312x} -1170 \color{blue}{+312x} \color{blue}{-90} \\\Leftrightarrow & 377x& = & -1260 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-1260}{377} \\\Leftrightarrow & x = \frac{-1260}{377} & & \\ & V = \left\{ \frac{-1260}{377} \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{-7}{4}x-2 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-63}{ \color{blue}{36} }x-\frac{72}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & -63x-72 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{+63x} & = & \color{red}{-63x} -72 \color{blue}{+63x} \color{blue}{-8} \\\Leftrightarrow & 75x& = & -80 \\\Leftrightarrow & \frac{75x}{ \color{red}{75} }& = & \frac{-80}{75} \\\Leftrightarrow & x = \frac{-16}{15} & & \\ & V = \left\{ \frac{-16}{15} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 45x-270 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-45x} & = & \color{red}{45x} -270 \color{blue}{-45x} \color{blue}{-40} \\\Leftrightarrow & -27x& = & -310 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-310}{-27} \\\Leftrightarrow & x = \frac{310}{27} & & \\ & V = \left\{ \frac{310}{27} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 33x-528 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-33x} & = & \color{red}{33x} -528 \color{blue}{-33x} \color{blue}{-30} \\\Leftrightarrow & -11x& = & -558 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-558}{-11} \\\Leftrightarrow & x = \frac{558}{11} & & \\ & V = \left\{ \frac{558}{11} \right\} & \\\end{align}\)
12. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & 154x+330 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{-154x} & = & \color{red}{154x} +330 \color{blue}{-154x} \color{blue}{-12} \\\Leftrightarrow & -121x& = & 318 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{318}{-121} \\\Leftrightarrow & x = \frac{-318}{121} & & \\ & V = \left\{ \frac{-318}{121} \right\} & \\\end{align}\)