Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{5}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{70}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 70x-294 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-70x} & = & \color{red}{70x} -294 \color{blue}{-70x} \color{blue}{+24} \\\Leftrightarrow & -49x& = & -270 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-270}{-49} \\\Leftrightarrow & x = \frac{270}{49} & & \\ & V = \left\{ \frac{270}{49} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 39x+468 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-39x} & = & \color{red}{39x} +468 \color{blue}{-39x} \color{blue}{-12} \\\Leftrightarrow & -13x& = & 456 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{456}{-13} \\\Leftrightarrow & x = \frac{-456}{13} & & \\ & V = \left\{ \frac{-456}{13} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{7}{5}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{98}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-30& = & 98x+280 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{-98x} & = & \color{red}{98x} +280 \color{blue}{-98x} \color{blue}{+30} \\\Leftrightarrow & -63x& = & 310 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{310}{-63} \\\Leftrightarrow & x = \frac{-310}{63} & & \\ & V = \left\{ \frac{-310}{63} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{156}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & 156x-130 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{-156x} & = & \color{red}{156x} -130 \color{blue}{-156x} \color{blue}{-20} \\\Leftrightarrow & -91x& = & -150 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-150}{-91} \\\Leftrightarrow & x = \frac{150}{91} & & \\ & V = \left\{ \frac{150}{91} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{2}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 84x+630 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-84x} & = & \color{red}{84x} +630 \color{blue}{-84x} \color{blue}{+120} \\\Leftrightarrow & -49x& = & 750 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{750}{-49} \\\Leftrightarrow & x = \frac{-750}{49} & & \\ & V = \left\{ \frac{-750}{49} \right\} & \\\end{align}\)
6. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{672}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 56x-672 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-56x} & = & \color{red}{56x} -672 \color{blue}{-56x} \color{blue}{-35} \\\Leftrightarrow & -40x& = & -707 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-707}{-40} \\\Leftrightarrow & x = \frac{707}{40} & & \\ & V = \left\{ \frac{707}{40} \right\} & \\\end{align}\)
7. \(\text{40 is het kleinste gemene veelvoud van 2, 8 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{8}& = & \frac{2}{5}x-6 \\\Leftrightarrow & \color{blue}{40.} (\frac{20x}{ \color{blue}{40} }- \frac{ 15 }{ \color{blue}{40} })& = & (\frac{16}{ \color{blue}{40} }x-\frac{240}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 20x-15& = & 16x-240 \\\Leftrightarrow & 20x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} -240 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & 4x& = & -225 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-225}{4} \\\Leftrightarrow & x = \frac{-225}{4} & & \\ & V = \left\{ \frac{-225}{4} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & -104x+624 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{+104x} & = & \color{red}{-104x} +624 \color{blue}{+104x} \color{blue}{+36} \\\Leftrightarrow & 143x& = & 660 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{660}{143} \\\Leftrightarrow & x = \frac{60}{13} & & \\ & V = \left\{ \frac{60}{13} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & -56x-350 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{+56x} & = & \color{red}{-56x} -350 \color{blue}{+56x} \color{blue}{-20} \\\Leftrightarrow & 91x& = & -370 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-370}{91} \\\Leftrightarrow & x = \frac{-370}{91} & & \\ & V = \left\{ \frac{-370}{91} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 6x+90 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} +90 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & -x& = & 86 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{86}{-1} \\\Leftrightarrow & x = -86 & & \\ & V = \left\{ -86 \right\} & \\\end{align}\)
11. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x-\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & 16x-320 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{-16x} & = & \color{red}{16x} -320 \color{blue}{-16x} \color{blue}{+25} \\\Leftrightarrow & 24x& = & -295 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{-295}{24} \\\Leftrightarrow & x = \frac{-295}{24} & & \\ & V = \left\{ \frac{-295}{24} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 28x-336 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-28x} & = & \color{red}{28x} -336 \color{blue}{-28x} \color{blue}{+36} \\\Leftrightarrow & -7x& = & -300 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-300}{-7} \\\Leftrightarrow & x = \frac{300}{7} & & \\ & V = \left\{ \frac{300}{7} \right\} & \\\end{align}\)