Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{3}{5}x-8 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{84}{ \color{blue}{140} }x-\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 84x-1120 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-84x} & = & \color{red}{84x} -1120 \color{blue}{-84x} \color{blue}{-40} \\\Leftrightarrow & -49x& = & -1160 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-1160}{-49} \\\Leftrightarrow & x = \frac{1160}{49} & & \\ & V = \left\{ \frac{1160}{49} \right\} & \\\end{align}\)
2. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }- \frac{ 14 }{ \color{blue}{63} })& = & (\frac{-105}{ \color{blue}{63} }x-\frac{441}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x-14& = & -105x-441 \\\Leftrightarrow & 9x \color{red}{-14} \color{blue}{+14} \color{blue}{+105x} & = & \color{red}{-105x} -441 \color{blue}{+105x} \color{blue}{+14} \\\Leftrightarrow & 114x& = & -427 \\\Leftrightarrow & \frac{114x}{ \color{red}{114} }& = & \frac{-427}{114} \\\Leftrightarrow & x = \frac{-427}{114} & & \\ & V = \left\{ \frac{-427}{114} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -546x-1950 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+546x} & = & \color{red}{-546x} -1950 \color{blue}{+546x} \color{blue}{-120} \\\Leftrightarrow & 611x& = & -2070 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{-2070}{611} \\\Leftrightarrow & x = \frac{-2070}{611} & & \\ & V = \left\{ \frac{-2070}{611} \right\} & \\\end{align}\)
4. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 70 }{ \color{blue}{455} })& = & (\frac{-364}{ \color{blue}{455} }x-\frac{3185}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+70& = & -364x-3185 \\\Leftrightarrow & 65x \color{red}{+70} \color{blue}{-70} \color{blue}{+364x} & = & \color{red}{-364x} -3185 \color{blue}{+364x} \color{blue}{-70} \\\Leftrightarrow & 429x& = & -3255 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-3255}{429} \\\Leftrightarrow & x = \frac{-1085}{143} & & \\ & V = \left\{ \frac{-1085}{143} \right\} & \\\end{align}\)
5. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-3}{4}x-3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 16 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x-\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-16& = & -21x-84 \\\Leftrightarrow & 4x \color{red}{-16} \color{blue}{+16} \color{blue}{+21x} & = & \color{red}{-21x} -84 \color{blue}{+21x} \color{blue}{+16} \\\Leftrightarrow & 25x& = & -68 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{-68}{25} \\\Leftrightarrow & x = \frac{-68}{25} & & \\ & V = \left\{ \frac{-68}{25} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & -140x-336 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{+140x} & = & \color{red}{-140x} -336 \color{blue}{+140x} \color{blue}{+35} \\\Leftrightarrow & 152x& = & -301 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{-301}{152} \\\Leftrightarrow & x = \frac{-301}{152} & & \\ & V = \left\{ \frac{-301}{152} \right\} & \\\end{align}\)
7. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{4}{5}x+3 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{8}{ \color{blue}{10} }x+\frac{30}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x-3& = & 8x+30 \\\Leftrightarrow & 5x \color{red}{-3} \color{blue}{+3} \color{blue}{-8x} & = & \color{red}{8x} +30 \color{blue}{-8x} \color{blue}{+3} \\\Leftrightarrow & -3x& = & 33 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{33}{-3} \\\Leftrightarrow & x = -11 & & \\ & V = \left\{ -11 \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{9}& = & \frac{-5}{6}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-50& = & -75x+180 \\\Leftrightarrow & 18x \color{red}{-50} \color{blue}{+50} \color{blue}{+75x} & = & \color{red}{-75x} +180 \color{blue}{+75x} \color{blue}{+50} \\\Leftrightarrow & 93x& = & 230 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{230}{93} \\\Leftrightarrow & x = \frac{230}{93} & & \\ & V = \left\{ \frac{230}{93} \right\} & \\\end{align}\)
9. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 175 }{ \color{blue}{385} })& = & (\frac{77}{ \color{blue}{385} }x+\frac{2310}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-175& = & 77x+2310 \\\Leftrightarrow & 55x \color{red}{-175} \color{blue}{+175} \color{blue}{-77x} & = & \color{red}{77x} +2310 \color{blue}{-77x} \color{blue}{+175} \\\Leftrightarrow & -22x& = & 2485 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{2485}{-22} \\\Leftrightarrow & x = \frac{-2485}{22} & & \\ & V = \left\{ \frac{-2485}{22} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+30& = & -56x-140 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{+56x} & = & \color{red}{-56x} -140 \color{blue}{+56x} \color{blue}{-30} \\\Leftrightarrow & 91x& = & -170 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-170}{91} \\\Leftrightarrow & x = \frac{-170}{91} & & \\ & V = \left\{ \frac{-170}{91} \right\} & \\\end{align}\)
11. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x+\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+15& = & 16x+240 \\\Leftrightarrow & 40x \color{red}{+15} \color{blue}{-15} \color{blue}{-16x} & = & \color{red}{16x} +240 \color{blue}{-16x} \color{blue}{-15} \\\Leftrightarrow & 24x& = & 225 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{225}{24} \\\Leftrightarrow & x = \frac{75}{8} & & \\ & V = \left\{ \frac{75}{8} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{112}{ \color{blue}{80} }x+\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & 112x+80 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{-112x} & = & \color{red}{112x} +80 \color{blue}{-112x} \color{blue}{-15} \\\Leftrightarrow & -92x& = & 65 \\\Leftrightarrow & \frac{-92x}{ \color{red}{-92} }& = & \frac{65}{-92} \\\Leftrightarrow & x = \frac{-65}{92} & & \\ & V = \left\{ \frac{-65}{92} \right\} & \\\end{align}\)