Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{112}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & 56x-112 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{-56x} & = & \color{red}{56x} -112 \color{blue}{-56x} \color{blue}{+35} \\\Leftrightarrow & -40x& = & -77 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-77}{-40} \\\Leftrightarrow & x = \frac{77}{40} & & \\ & V = \left\{ \frac{77}{40} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{2}{5}x+7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{156}{ \color{blue}{390} }x+\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 156x+2730 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-156x} & = & \color{red}{156x} +2730 \color{blue}{-156x} \color{blue}{+60} \\\Leftrightarrow & -91x& = & 2790 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{2790}{-91} \\\Leftrightarrow & x = \frac{-2790}{91} & & \\ & V = \left\{ \frac{-2790}{91} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x+\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & -60x+144 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+60x} & = & \color{red}{-60x} +144 \color{blue}{+60x} \color{blue}{-8} \\\Leftrightarrow & 69x& = & 136 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{136}{69} \\\Leftrightarrow & x = \frac{136}{69} & & \\ & V = \left\{ \frac{136}{69} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & -294x-1050 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{+294x} & = & \color{red}{-294x} -1050 \color{blue}{+294x} \color{blue}{-75} \\\Leftrightarrow & 329x& = & -1125 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-1125}{329} \\\Leftrightarrow & x = \frac{-1125}{329} & & \\ & V = \left\{ \frac{-1125}{329} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+25& = & 35x+70 \\\Leftrightarrow & 14x \color{red}{+25} \color{blue}{-25} \color{blue}{-35x} & = & \color{red}{35x} +70 \color{blue}{-35x} \color{blue}{-25} \\\Leftrightarrow & -21x& = & 45 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{45}{-21} \\\Leftrightarrow & x = \frac{-15}{7} & & \\ & V = \left\{ \frac{-15}{7} \right\} & \\\end{align}\)
6. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{3}{2}x-4 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{15}{ \color{blue}{10} }x-\frac{40}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x+3& = & 15x-40 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-15x} & = & \color{red}{15x} -40 \color{blue}{-15x} \color{blue}{-3} \\\Leftrightarrow & -13x& = & -43 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-43}{-13} \\\Leftrightarrow & x = \frac{43}{13} & & \\ & V = \left\{ \frac{43}{13} \right\} & \\\end{align}\)
7. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 56 }{ \color{blue}{182} })& = & (\frac{273}{ \color{blue}{182} }x-\frac{546}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+56& = & 273x-546 \\\Leftrightarrow & 26x \color{red}{+56} \color{blue}{-56} \color{blue}{-273x} & = & \color{red}{273x} -546 \color{blue}{-273x} \color{blue}{-56} \\\Leftrightarrow & -247x& = & -602 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-602}{-247} \\\Leftrightarrow & x = \frac{602}{247} & & \\ & V = \left\{ \frac{602}{247} \right\} & \\\end{align}\)
8. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{3}{5}x+6 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 100 }{ \color{blue}{180} })& = & (\frac{108}{ \color{blue}{180} }x+\frac{1080}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-100& = & 108x+1080 \\\Leftrightarrow & 45x \color{red}{-100} \color{blue}{+100} \color{blue}{-108x} & = & \color{red}{108x} +1080 \color{blue}{-108x} \color{blue}{+100} \\\Leftrightarrow & -63x& = & 1180 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{1180}{-63} \\\Leftrightarrow & x = \frac{-1180}{63} & & \\ & V = \left\{ \frac{-1180}{63} \right\} & \\\end{align}\)
9. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 175 }{ \color{blue}{385} })& = & (\frac{77}{ \color{blue}{385} }x-\frac{2695}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-175& = & 77x-2695 \\\Leftrightarrow & 55x \color{red}{-175} \color{blue}{+175} \color{blue}{-77x} & = & \color{red}{77x} -2695 \color{blue}{-77x} \color{blue}{+175} \\\Leftrightarrow & -22x& = & -2520 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{-2520}{-22} \\\Leftrightarrow & x = \frac{1260}{11} & & \\ & V = \left\{ \frac{1260}{11} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -28x-168 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+28x} & = & \color{red}{-28x} -168 \color{blue}{+28x} \color{blue}{+12} \\\Leftrightarrow & 49x& = & -156 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-156}{49} \\\Leftrightarrow & x = \frac{-156}{49} & & \\ & V = \left\{ \frac{-156}{49} \right\} & \\\end{align}\)
11. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }- \frac{ 4 }{ \color{blue}{15} })& = & (\frac{20}{ \color{blue}{15} }x+\frac{120}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x-4& = & 20x+120 \\\Leftrightarrow & 3x \color{red}{-4} \color{blue}{+4} \color{blue}{-20x} & = & \color{red}{20x} +120 \color{blue}{-20x} \color{blue}{+4} \\\Leftrightarrow & -17x& = & 124 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{124}{-17} \\\Leftrightarrow & x = \frac{-124}{17} & & \\ & V = \left\{ \frac{-124}{17} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{22}{ \color{blue}{110} }x+\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-40& = & 22x+770 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-22x} & = & \color{red}{22x} +770 \color{blue}{-22x} \color{blue}{+40} \\\Leftrightarrow & 33x& = & 810 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{810}{33} \\\Leftrightarrow & x = \frac{270}{11} & & \\ & V = \left\{ \frac{270}{11} \right\} & \\\end{align}\)