Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-7}{4}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+80& = & -245x-980 \\\Leftrightarrow & 28x \color{red}{+80} \color{blue}{-80} \color{blue}{+245x} & = & \color{red}{-245x} -980 \color{blue}{+245x} \color{blue}{-80} \\\Leftrightarrow & 273x& = & -1060 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{-1060}{273} \\\Leftrightarrow & x = \frac{-1060}{273} & & \\ & V = \left\{ \frac{-1060}{273} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{260}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 260x+156 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-260x} & = & \color{red}{260x} +156 \color{blue}{-260x} \color{blue}{-48} \\\Leftrightarrow & -221x& = & 108 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{108}{-221} \\\Leftrightarrow & x = \frac{-108}{221} & & \\ & V = \left\{ \frac{-108}{221} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{5}{4}x-3 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{45}{ \color{blue}{36} }x-\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & 45x-108 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{-45x} & = & \color{red}{45x} -108 \color{blue}{-45x} \color{blue}{-8} \\\Leftrightarrow & -33x& = & -116 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-116}{-33} \\\Leftrightarrow & x = \frac{116}{33} & & \\ & V = \left\{ \frac{116}{33} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{1}{4}x-8 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x-\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+36& = & 39x-1248 \\\Leftrightarrow & 52x \color{red}{+36} \color{blue}{-36} \color{blue}{-39x} & = & \color{red}{39x} -1248 \color{blue}{-39x} \color{blue}{-36} \\\Leftrightarrow & 13x& = & -1284 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-1284}{13} \\\Leftrightarrow & x = \frac{-1284}{13} & & \\ & V = \left\{ \frac{-1284}{13} \right\} & \\\end{align}\)
5. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{-7}{4}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+40& = & -245x+840 \\\Leftrightarrow & 28x \color{red}{+40} \color{blue}{-40} \color{blue}{+245x} & = & \color{red}{-245x} +840 \color{blue}{+245x} \color{blue}{-40} \\\Leftrightarrow & 273x& = & 800 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{800}{273} \\\Leftrightarrow & x = \frac{800}{273} & & \\ & V = \left\{ \frac{800}{273} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{320}{ \color{blue}{240} }x-\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & 320x-720 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{-320x} & = & \color{red}{320x} -720 \color{blue}{-320x} \color{blue}{-75} \\\Leftrightarrow & -272x& = & -795 \\\Leftrightarrow & \frac{-272x}{ \color{red}{-272} }& = & \frac{-795}{-272} \\\Leftrightarrow & x = \frac{795}{272} & & \\ & V = \left\{ \frac{795}{272} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & 176x-396 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-176x} & = & \color{red}{176x} -396 \color{blue}{-176x} \color{blue}{-24} \\\Leftrightarrow & -143x& = & -420 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-420}{-143} \\\Leftrightarrow & x = \frac{420}{143} & & \\ & V = \left\{ \frac{420}{143} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 5, 14 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{5}{4}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 30 }{ \color{blue}{140} })& = & (\frac{175}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-30& = & 175x-980 \\\Leftrightarrow & 28x \color{red}{-30} \color{blue}{+30} \color{blue}{-175x} & = & \color{red}{175x} -980 \color{blue}{-175x} \color{blue}{+30} \\\Leftrightarrow & -147x& = & -950 \\\Leftrightarrow & \frac{-147x}{ \color{red}{-147} }& = & \frac{-950}{-147} \\\Leftrightarrow & x = \frac{950}{147} & & \\ & V = \left\{ \frac{950}{147} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{1}{4}x-7 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{12}{ \color{blue}{48} }x-\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & 12x-336 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-12x} & = & \color{red}{12x} -336 \color{blue}{-12x} \color{blue}{+15} \\\Leftrightarrow & 4x& = & -321 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-321}{4} \\\Leftrightarrow & x = \frac{-321}{4} & & \\ & V = \left\{ \frac{-321}{4} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-9& = & -70x+42 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{+70x} & = & \color{red}{-70x} +42 \color{blue}{+70x} \color{blue}{+9} \\\Leftrightarrow & 76x& = & 51 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{51}{76} \\\Leftrightarrow & x = \frac{51}{76} & & \\ & V = \left\{ \frac{51}{76} \right\} & \\\end{align}\)
11. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 7x+70 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-7x} & = & \color{red}{7x} +70 \color{blue}{-7x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 74 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{74}{-5} \\\Leftrightarrow & x = \frac{-74}{5} & & \\ & V = \left\{ \frac{-74}{5} \right\} & \\\end{align}\)
12. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{52}{ \color{blue}{260} }x-\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+40& = & 52x-780 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{-52x} & = & \color{red}{52x} -780 \color{blue}{-52x} \color{blue}{-40} \\\Leftrightarrow & 13x& = & -820 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-820}{13} \\\Leftrightarrow & x = \frac{-820}{13} & & \\ & V = \left\{ \frac{-820}{13} \right\} & \\\end{align}\)