Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 15x+240 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} +240 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -9x& = & 248 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{248}{-9} \\\Leftrightarrow & x = \frac{-248}{9} & & \\ & V = \left\{ \frac{-248}{9} \right\} & \\\end{align}\)
2. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 30 }{ \color{blue}{195} })& = & (\frac{-325}{ \color{blue}{195} }x+\frac{975}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-30& = & -325x+975 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{+325x} & = & \color{red}{-325x} +975 \color{blue}{+325x} \color{blue}{+30} \\\Leftrightarrow & 364x& = & 1005 \\\Leftrightarrow & \frac{364x}{ \color{red}{364} }& = & \frac{1005}{364} \\\Leftrightarrow & x = \frac{1005}{364} & & \\ & V = \left\{ \frac{1005}{364} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{-231}{ \color{blue}{165} }x+\frac{1320}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+75& = & -231x+1320 \\\Leftrightarrow & 55x \color{red}{+75} \color{blue}{-75} \color{blue}{+231x} & = & \color{red}{-231x} +1320 \color{blue}{+231x} \color{blue}{-75} \\\Leftrightarrow & 286x& = & 1245 \\\Leftrightarrow & \frac{286x}{ \color{red}{286} }& = & \frac{1245}{286} \\\Leftrightarrow & x = \frac{1245}{286} & & \\ & V = \left\{ \frac{1245}{286} \right\} & \\\end{align}\)
4. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 105 }{ \color{blue}{336} })& = & (\frac{560}{ \color{blue}{336} }x+\frac{2352}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+105& = & 560x+2352 \\\Leftrightarrow & 48x \color{red}{+105} \color{blue}{-105} \color{blue}{-560x} & = & \color{red}{560x} +2352 \color{blue}{-560x} \color{blue}{-105} \\\Leftrightarrow & -512x& = & 2247 \\\Leftrightarrow & \frac{-512x}{ \color{red}{-512} }& = & \frac{2247}{-512} \\\Leftrightarrow & x = \frac{-2247}{512} & & \\ & V = \left\{ \frac{-2247}{512} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -70x+126 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+70x} & = & \color{red}{-70x} +126 \color{blue}{+70x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & 108 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{108}{91} \\\Leftrightarrow & x = \frac{108}{91} & & \\ & V = \left\{ \frac{108}{91} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 126x-1260 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-126x} & = & \color{red}{126x} -1260 \color{blue}{-126x} \color{blue}{+120} \\\Leftrightarrow & -91x& = & -1140 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-1140}{-91} \\\Leftrightarrow & x = \frac{1140}{91} & & \\ & V = \left\{ \frac{1140}{91} \right\} & \\\end{align}\)
7. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{6}{ \color{blue}{12} }x-\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & 6x-72 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{-6x} & = & \color{red}{6x} -72 \color{blue}{-6x} \color{blue}{-5} \\\Leftrightarrow & -2x& = & -77 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-77}{-2} \\\Leftrightarrow & x = \frac{77}{2} & & \\ & V = \left\{ \frac{77}{2} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 22x-264 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-22x} & = & \color{red}{22x} -264 \color{blue}{-22x} \color{blue}{+12} \\\Leftrightarrow & 11x& = & -252 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-252}{11} \\\Leftrightarrow & x = \frac{-252}{11} & & \\ & V = \left\{ \frac{-252}{11} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x+126 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} +126 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 150 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{150}{-7} \\\Leftrightarrow & x = \frac{-150}{7} & & \\ & V = \left\{ \frac{-150}{7} \right\} & \\\end{align}\)
10. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{7}{5}x-5 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 80 }{ \color{blue}{260} })& = & (\frac{364}{ \color{blue}{260} }x-\frac{1300}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+80& = & 364x-1300 \\\Leftrightarrow & 65x \color{red}{+80} \color{blue}{-80} \color{blue}{-364x} & = & \color{red}{364x} -1300 \color{blue}{-364x} \color{blue}{-80} \\\Leftrightarrow & -299x& = & -1380 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{-1380}{-299} \\\Leftrightarrow & x = \frac{60}{13} & & \\ & V = \left\{ \frac{60}{13} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{5}{3}x-3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{260}{ \color{blue}{156} }x-\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+36& = & 260x-468 \\\Leftrightarrow & 39x \color{red}{+36} \color{blue}{-36} \color{blue}{-260x} & = & \color{red}{260x} -468 \color{blue}{-260x} \color{blue}{-36} \\\Leftrightarrow & -221x& = & -504 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{-504}{-221} \\\Leftrightarrow & x = \frac{504}{221} & & \\ & V = \left\{ \frac{504}{221} \right\} & \\\end{align}\)
12. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-96}{ \color{blue}{120} }x-\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & -96x-600 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{+96x} & = & \color{red}{-96x} -600 \color{blue}{+96x} \color{blue}{+75} \\\Leftrightarrow & 116x& = & -525 \\\Leftrightarrow & \frac{116x}{ \color{red}{116} }& = & \frac{-525}{116} \\\Leftrightarrow & x = \frac{-525}{116} & & \\ & V = \left\{ \frac{-525}{116} \right\} & \\\end{align}\)