Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}+\frac{3}{13}=\frac{1}{2}x+4\)
2. \(\frac{x}{5}+\frac{4}{9}=\frac{-5}{6}x+5\)
3. \(\frac{x}{2}+\frac{5}{7}=\frac{3}{5}x-5\)
4. \(\frac{x}{7}+\frac{2}{15}=\frac{7}{6}x-8\)
5. \(\frac{x}{4}-\frac{4}{9}=\frac{1}{3}x-5\)
6. \(\frac{x}{2}-\frac{4}{9}=\frac{-4}{5}x-1\)
7. \(\frac{x}{5}-\frac{4}{7}=\frac{1}{2}x-3\)
8. \(\frac{x}{2}+\frac{2}{9}=\frac{-4}{5}x-6\)
9. \(\frac{x}{4}+\frac{4}{13}=\frac{2}{3}x+2\)
10. \(\frac{x}{6}+\frac{5}{11}=\frac{1}{5}x-3\)
11. \(\frac{x}{7}-\frac{5}{14}=\frac{1}{2}x-6\)
12. \(\frac{x}{6}+\frac{5}{16}=\frac{3}{5}x+2\)

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x+\frac{520}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x+520 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} +520 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & 490 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{490}{-39} \\\Leftrightarrow & x = \frac{-490}{39} & & \\ & V = \left\{ \frac{-490}{39} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{-5}{6}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & -75x+450 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{+75x} & = & \color{red}{-75x} +450 \color{blue}{+75x} \color{blue}{-40} \\\Leftrightarrow & 93x& = & 410 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{410}{93} \\\Leftrightarrow & x = \frac{410}{93} & & \\ & V = \left\{ \frac{410}{93} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{3}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 50 }{ \color{blue}{70} })& = & (\frac{42}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+50& = & 42x-350 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{-42x} & = & \color{red}{42x} -350 \color{blue}{-42x} \color{blue}{-50} \\\Leftrightarrow & -7x& = & -400 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-400}{-7} \\\Leftrightarrow & x = \frac{400}{7} & & \\ & V = \left\{ \frac{400}{7} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{7}{6}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & 245x-1680 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{-245x} & = & \color{red}{245x} -1680 \color{blue}{-245x} \color{blue}{-28} \\\Leftrightarrow & -215x& = & -1708 \\\Leftrightarrow & \frac{-215x}{ \color{red}{-215} }& = & \frac{-1708}{-215} \\\Leftrightarrow & x = \frac{1708}{215} & & \\ & V = \left\{ \frac{1708}{215} \right\} & \\\end{align}\)
5. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x-\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & 12x-180 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{-12x} & = & \color{red}{12x} -180 \color{blue}{-12x} \color{blue}{+16} \\\Leftrightarrow & -3x& = & -164 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-164}{-3} \\\Leftrightarrow & x = \frac{164}{3} & & \\ & V = \left\{ \frac{164}{3} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-40& = & -72x-90 \\\Leftrightarrow & 45x \color{red}{-40} \color{blue}{+40} \color{blue}{+72x} & = & \color{red}{-72x} -90 \color{blue}{+72x} \color{blue}{+40} \\\Leftrightarrow & 117x& = & -50 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-50}{117} \\\Leftrightarrow & x = \frac{-50}{117} & & \\ & V = \left\{ \frac{-50}{117} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 35x-210 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-35x} & = & \color{red}{35x} -210 \color{blue}{-35x} \color{blue}{+40} \\\Leftrightarrow & -21x& = & -170 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-170}{-21} \\\Leftrightarrow & x = \frac{170}{21} & & \\ & V = \left\{ \frac{170}{21} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+20& = & -72x-540 \\\Leftrightarrow & 45x \color{red}{+20} \color{blue}{-20} \color{blue}{+72x} & = & \color{red}{-72x} -540 \color{blue}{+72x} \color{blue}{-20} \\\Leftrightarrow & 117x& = & -560 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-560}{117} \\\Leftrightarrow & x = \frac{-560}{117} & & \\ & V = \left\{ \frac{-560}{117} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{104}{ \color{blue}{156} }x+\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 104x+312 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-104x} & = & \color{red}{104x} +312 \color{blue}{-104x} \color{blue}{-48} \\\Leftrightarrow & -65x& = & 264 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{264}{-65} \\\Leftrightarrow & x = \frac{-264}{65} & & \\ & V = \left\{ \frac{-264}{65} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & 66x-990 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{-66x} & = & \color{red}{66x} -990 \color{blue}{-66x} \color{blue}{-150} \\\Leftrightarrow & -11x& = & -1140 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-1140}{-11} \\\Leftrightarrow & x = \frac{1140}{11} & & \\ & V = \left\{ \frac{1140}{11} \right\} & \\\end{align}\)
11. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{14}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 5 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-5& = & 7x-84 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-7x} & = & \color{red}{7x} -84 \color{blue}{-7x} \color{blue}{+5} \\\Leftrightarrow & -5x& = & -79 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-79}{-5} \\\Leftrightarrow & x = \frac{79}{5} & & \\ & V = \left\{ \frac{79}{5} \right\} & \\\end{align}\)
12. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{144}{ \color{blue}{240} }x+\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & 144x+480 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-144x} & = & \color{red}{144x} +480 \color{blue}{-144x} \color{blue}{-75} \\\Leftrightarrow & -104x& = & 405 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{405}{-104} \\\Leftrightarrow & x = \frac{-405}{104} & & \\ & V = \left\{ \frac{-405}{104} \right\} & \\\end{align}\)