Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -168x-1260 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+168x} & = & \color{red}{-168x} -1260 \color{blue}{+168x} \color{blue}{-120} \\\Leftrightarrow & 203x& = & -1380 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1380}{203} \\\Leftrightarrow & x = \frac{-1380}{203} & & \\ & V = \left\{ \frac{-1380}{203} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{104}{ \color{blue}{130} }x+\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-20& = & 104x+260 \\\Leftrightarrow & 65x \color{red}{-20} \color{blue}{+20} \color{blue}{-104x} & = & \color{red}{104x} +260 \color{blue}{-104x} \color{blue}{+20} \\\Leftrightarrow & -39x& = & 280 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{280}{-39} \\\Leftrightarrow & x = \frac{-280}{39} & & \\ & V = \left\{ \frac{-280}{39} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{364}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 364x-780 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-364x} & = & \color{red}{364x} -780 \color{blue}{-364x} \color{blue}{-48} \\\Leftrightarrow & -325x& = & -828 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{-828}{-325} \\\Leftrightarrow & x = \frac{828}{325} & & \\ & V = \left\{ \frac{828}{325} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{7}{6}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{105}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 105x-720 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-105x} & = & \color{red}{105x} -720 \color{blue}{-105x} \color{blue}{+20} \\\Leftrightarrow & -87x& = & -700 \\\Leftrightarrow & \frac{-87x}{ \color{red}{-87} }& = & \frac{-700}{-87} \\\Leftrightarrow & x = \frac{700}{87} & & \\ & V = \left\{ \frac{700}{87} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & -312x-2340 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{+312x} & = & \color{red}{-312x} -2340 \color{blue}{+312x} \color{blue}{+120} \\\Leftrightarrow & 377x& = & -2220 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-2220}{377} \\\Leftrightarrow & x = \frac{-2220}{377} & & \\ & V = \left\{ \frac{-2220}{377} \right\} & \\\end{align}\)
6. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{72}{ \color{blue}{120} }x+\frac{840}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & 72x+840 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{-72x} & = & \color{red}{72x} +840 \color{blue}{-72x} \color{blue}{-75} \\\Leftrightarrow & -52x& = & 765 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{765}{-52} \\\Leftrightarrow & x = \frac{-765}{52} & & \\ & V = \left\{ \frac{-765}{52} \right\} & \\\end{align}\)
7. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 12 }{ \color{blue}{21} })& = & (\frac{-35}{ \color{blue}{21} }x+\frac{105}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+12& = & -35x+105 \\\Leftrightarrow & 3x \color{red}{+12} \color{blue}{-12} \color{blue}{+35x} & = & \color{red}{-35x} +105 \color{blue}{+35x} \color{blue}{-12} \\\Leftrightarrow & 38x& = & 93 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{93}{38} \\\Leftrightarrow & x = \frac{93}{38} & & \\ & V = \left\{ \frac{93}{38} \right\} & \\\end{align}\)
8. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 56 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{182}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+56& = & 91x-182 \\\Leftrightarrow & 26x \color{red}{+56} \color{blue}{-56} \color{blue}{-91x} & = & \color{red}{91x} -182 \color{blue}{-91x} \color{blue}{-56} \\\Leftrightarrow & -65x& = & -238 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-238}{-65} \\\Leftrightarrow & x = \frac{238}{65} & & \\ & V = \left\{ \frac{238}{65} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 42x+630 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-42x} & = & \color{red}{42x} +630 \color{blue}{-42x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & 690 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{690}{-7} \\\Leftrightarrow & x = \frac{-690}{7} & & \\ & V = \left\{ \frac{-690}{7} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -56x-504 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+56x} & = & \color{red}{-56x} -504 \color{blue}{+56x} \color{blue}{-48} \\\Leftrightarrow & 77x& = & -552 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-552}{77} \\\Leftrightarrow & x = \frac{-552}{77} & & \\ & V = \left\{ \frac{-552}{77} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x+\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-60& = & 33x+528 \\\Leftrightarrow & 44x \color{red}{-60} \color{blue}{+60} \color{blue}{-33x} & = & \color{red}{33x} +528 \color{blue}{-33x} \color{blue}{+60} \\\Leftrightarrow & 11x& = & 588 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{588}{11} \\\Leftrightarrow & x = \frac{588}{11} & & \\ & V = \left\{ \frac{588}{11} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 65x-260 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-65x} & = & \color{red}{65x} -260 \color{blue}{-65x} \color{blue}{-40} \\\Leftrightarrow & -39x& = & -300 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-300}{-39} \\\Leftrightarrow & x = \frac{100}{13} & & \\ & V = \left\{ \frac{100}{13} \right\} & \\\end{align}\)