Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{5}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{70}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 70x+168 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-70x} & = & \color{red}{70x} +168 \color{blue}{-70x} \color{blue}{+12} \\\Leftrightarrow & -49x& = & 180 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{180}{-49} \\\Leftrightarrow & x = \frac{-180}{49} & & \\ & V = \left\{ \frac{-180}{49} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & 44x-132 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-44x} & = & \color{red}{44x} -132 \color{blue}{-44x} \color{blue}{-60} \\\Leftrightarrow & -11x& = & -192 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-192}{-11} \\\Leftrightarrow & x = \frac{192}{11} & & \\ & V = \left\{ \frac{192}{11} \right\} & \\\end{align}\)
3. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-352}{ \color{blue}{132} }x-\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -352x-132 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+352x} & = & \color{red}{-352x} -132 \color{blue}{+352x} \color{blue}{-24} \\\Leftrightarrow & 385x& = & -156 \\\Leftrightarrow & \frac{385x}{ \color{red}{385} }& = & \frac{-156}{385} \\\Leftrightarrow & x = \frac{-156}{385} & & \\ & V = \left\{ \frac{-156}{385} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 21 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+21& = & 35x+70 \\\Leftrightarrow & 10x \color{red}{+21} \color{blue}{-21} \color{blue}{-35x} & = & \color{red}{35x} +70 \color{blue}{-35x} \color{blue}{-21} \\\Leftrightarrow & -25x& = & 49 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{49}{-25} \\\Leftrightarrow & x = \frac{-49}{25} & & \\ & V = \left\{ \frac{-49}{25} \right\} & \\\end{align}\)
5. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-208}{ \color{blue}{78} }x+\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & -208x+312 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{+208x} & = & \color{red}{-208x} +312 \color{blue}{+208x} \color{blue}{-12} \\\Leftrightarrow & 247x& = & 300 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{300}{247} \\\Leftrightarrow & x = \frac{300}{247} & & \\ & V = \left\{ \frac{300}{247} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -72x+720 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+72x} & = & \color{red}{-72x} +720 \color{blue}{+72x} \color{blue}{-20} \\\Leftrightarrow & 87x& = & 700 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{700}{87} \\\Leftrightarrow & x = \frac{700}{87} & & \\ & V = \left\{ \frac{700}{87} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{3}{5}x-8 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{42}{ \color{blue}{70} }x-\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & 42x-560 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-42x} & = & \color{red}{42x} -560 \color{blue}{-42x} \color{blue}{+40} \\\Leftrightarrow & -7x& = & -520 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-520}{-7} \\\Leftrightarrow & x = \frac{520}{7} & & \\ & V = \left\{ \frac{520}{7} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{3}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 126x-1470 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-126x} & = & \color{red}{126x} -1470 \color{blue}{-126x} \color{blue}{+120} \\\Leftrightarrow & -91x& = & -1350 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-1350}{-91} \\\Leftrightarrow & x = \frac{1350}{91} & & \\ & V = \left\{ \frac{1350}{91} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{364}{ \color{blue}{156} }x-\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+36& = & 364x-1092 \\\Leftrightarrow & 39x \color{red}{+36} \color{blue}{-36} \color{blue}{-364x} & = & \color{red}{364x} -1092 \color{blue}{-364x} \color{blue}{-36} \\\Leftrightarrow & -325x& = & -1128 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{-1128}{-325} \\\Leftrightarrow & x = \frac{1128}{325} & & \\ & V = \left\{ \frac{1128}{325} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{5}{6}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{50}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & 50x-480 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{-50x} & = & \color{red}{50x} -480 \color{blue}{-50x} \color{blue}{-25} \\\Leftrightarrow & -38x& = & -505 \\\Leftrightarrow & \frac{-38x}{ \color{red}{-38} }& = & \frac{-505}{-38} \\\Leftrightarrow & x = \frac{505}{38} & & \\ & V = \left\{ \frac{505}{38} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 30 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+30& = & -70x+210 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{+70x} & = & \color{red}{-70x} +210 \color{blue}{+70x} \color{blue}{-30} \\\Leftrightarrow & 91x& = & 180 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{180}{91} \\\Leftrightarrow & x = \frac{180}{91} & & \\ & V = \left\{ \frac{180}{91} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-30& = & 35x+350 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-35x} & = & \color{red}{35x} +350 \color{blue}{-35x} \color{blue}{+30} \\\Leftrightarrow & -21x& = & 380 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{380}{-21} \\\Leftrightarrow & x = \frac{-380}{21} & & \\ & V = \left\{ \frac{-380}{21} \right\} & \\\end{align}\)