Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{1}{4}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-18& = & 21x-588 \\\Leftrightarrow & 28x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} -588 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & 7x& = & -570 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-570}{7} \\\Leftrightarrow & x = \frac{-570}{7} & & \\ & V = \left\{ \frac{-570}{7} \right\} & \\\end{align}\)
2. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 105 }{ \color{blue}{455} })& = & (\frac{-182}{ \color{blue}{455} }x-\frac{3640}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+105& = & -182x-3640 \\\Leftrightarrow & 65x \color{red}{+105} \color{blue}{-105} \color{blue}{+182x} & = & \color{red}{-182x} -3640 \color{blue}{+182x} \color{blue}{-105} \\\Leftrightarrow & 247x& = & -3745 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-3745}{247} \\\Leftrightarrow & x = \frac{-3745}{247} & & \\ & V = \left\{ \frac{-3745}{247} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{2}{5}x+3 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{32}{ \color{blue}{80} }x+\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & 32x+240 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{-32x} & = & \color{red}{32x} +240 \color{blue}{-32x} \color{blue}{+25} \\\Leftrightarrow & -12x& = & 265 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{265}{-12} \\\Leftrightarrow & x = \frac{-265}{12} & & \\ & V = \left\{ \frac{-265}{12} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x-\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-10& = & 6x-108 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{-6x} & = & \color{red}{6x} -108 \color{blue}{-6x} \color{blue}{+10} \\\Leftrightarrow & 3x& = & -98 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-98}{3} \\\Leftrightarrow & x = \frac{-98}{3} & & \\ & V = \left\{ \frac{-98}{3} \right\} & \\\end{align}\)
5. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{5}{4}x-2 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }- \frac{ 112 }{ \color{blue}{252} })& = & (\frac{315}{ \color{blue}{252} }x-\frac{504}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x-112& = & 315x-504 \\\Leftrightarrow & 36x \color{red}{-112} \color{blue}{+112} \color{blue}{-315x} & = & \color{red}{315x} -504 \color{blue}{-315x} \color{blue}{+112} \\\Leftrightarrow & -279x& = & -392 \\\Leftrightarrow & \frac{-279x}{ \color{red}{-279} }& = & \frac{-392}{-279} \\\Leftrightarrow & x = \frac{392}{279} & & \\ & V = \left\{ \frac{392}{279} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 5, 6 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{-3}{4}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 50 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-50& = & -45x-180 \\\Leftrightarrow & 12x \color{red}{-50} \color{blue}{+50} \color{blue}{+45x} & = & \color{red}{-45x} -180 \color{blue}{+45x} \color{blue}{+50} \\\Leftrightarrow & 57x& = & -130 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-130}{57} \\\Leftrightarrow & x = \frac{-130}{57} & & \\ & V = \left\{ \frac{-130}{57} \right\} & \\\end{align}\)
7. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{9}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }+ \frac{ 25 }{ \color{blue}{45} })& = & (\frac{30}{ \color{blue}{45} }x+\frac{90}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x+25& = & 30x+90 \\\Leftrightarrow & 9x \color{red}{+25} \color{blue}{-25} \color{blue}{-30x} & = & \color{red}{30x} +90 \color{blue}{-30x} \color{blue}{-25} \\\Leftrightarrow & -21x& = & 65 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{65}{-21} \\\Leftrightarrow & x = \frac{-65}{21} & & \\ & V = \left\{ \frac{-65}{21} \right\} & \\\end{align}\)
8. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{2}{5}x+7 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{52}{ \color{blue}{130} }x+\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-20& = & 52x+910 \\\Leftrightarrow & 65x \color{red}{-20} \color{blue}{+20} \color{blue}{-52x} & = & \color{red}{52x} +910 \color{blue}{-52x} \color{blue}{+20} \\\Leftrightarrow & 13x& = & 930 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{930}{13} \\\Leftrightarrow & x = \frac{930}{13} & & \\ & V = \left\{ \frac{930}{13} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{3}{2}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 63x+84 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-63x} & = & \color{red}{63x} +84 \color{blue}{-63x} \color{blue}{-24} \\\Leftrightarrow & -49x& = & 60 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{60}{-49} \\\Leftrightarrow & x = \frac{-60}{49} & & \\ & V = \left\{ \frac{-60}{49} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{3}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{42}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & 42x-490 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{-42x} & = & \color{red}{42x} -490 \color{blue}{-42x} \color{blue}{+15} \\\Leftrightarrow & -7x& = & -475 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-475}{-7} \\\Leftrightarrow & x = \frac{475}{7} & & \\ & V = \left\{ \frac{475}{7} \right\} & \\\end{align}\)
11. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x-\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & -112x-320 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{+112x} & = & \color{red}{-112x} -320 \color{blue}{+112x} \color{blue}{-25} \\\Leftrightarrow & 152x& = & -345 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{-345}{152} \\\Leftrightarrow & x = \frac{-345}{152} & & \\ & V = \left\{ \frac{-345}{152} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x+\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & 44x+924 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-44x} & = & \color{red}{44x} +924 \color{blue}{-44x} \color{blue}{-60} \\\Leftrightarrow & -11x& = & 864 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{864}{-11} \\\Leftrightarrow & x = \frac{-864}{11} & & \\ & V = \left\{ \frac{-864}{11} \right\} & \\\end{align}\)