Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}+\frac{5}{8}=\frac{4}{5}x-4\)
2. \(\frac{x}{3}+\frac{2}{7}=\frac{1}{2}x+7\)
3. \(\frac{x}{7}+\frac{5}{16}=\frac{7}{4}x+6\)
4. \(\frac{x}{4}+\frac{4}{13}=\frac{-7}{5}x-2\)
5. \(\frac{x}{2}-\frac{2}{15}=\frac{-2}{5}x-2\)
6. \(\frac{x}{7}+\frac{3}{14}=\frac{3}{4}x+5\)
7. \(\frac{x}{3}-\frac{4}{11}=\frac{-4}{5}x+6\)
8. \(\frac{x}{4}-\frac{2}{7}=\frac{2}{3}x+7\)
9. \(\frac{x}{5}+\frac{4}{7}=\frac{1}{4}x-5\)
10. \(\frac{x}{4}-\frac{2}{9}=\frac{4}{3}x+1\)
11. \(\frac{x}{3}+\frac{2}{9}=\frac{1}{2}x+3\)
12. \(\frac{x}{6}-\frac{5}{16}=\frac{7}{5}x-8\)

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

Verbetersleutel

1. \(\text{120 is het kleinste gemene veelvoud van 3, 8 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{8}& = & \frac{4}{5}x-4 \\\Leftrightarrow & \color{blue}{120.} (\frac{40x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{96}{ \color{blue}{120} }x-\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 40x+75& = & 96x-480 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-96x} & = & \color{red}{96x} -480 \color{blue}{-96x} \color{blue}{-75} \\\Leftrightarrow & -56x& = & -555 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{-555}{-56} \\\Leftrightarrow & x = \frac{555}{56} & & \\ & V = \left\{ \frac{555}{56} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x+294 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} +294 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & 282 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{282}{-7} \\\Leftrightarrow & x = \frac{-282}{7} & & \\ & V = \left\{ \frac{-282}{7} \right\} & \\\end{align}\)
3. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{7}{4}x+6 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{196}{ \color{blue}{112} }x+\frac{672}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 196x+672 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-196x} & = & \color{red}{196x} +672 \color{blue}{-196x} \color{blue}{-35} \\\Leftrightarrow & -180x& = & 637 \\\Leftrightarrow & \frac{-180x}{ \color{red}{-180} }& = & \frac{637}{-180} \\\Leftrightarrow & x = \frac{-637}{180} & & \\ & V = \left\{ \frac{-637}{180} \right\} & \\\end{align}\)
4. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-364}{ \color{blue}{260} }x-\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+80& = & -364x-520 \\\Leftrightarrow & 65x \color{red}{+80} \color{blue}{-80} \color{blue}{+364x} & = & \color{red}{-364x} -520 \color{blue}{+364x} \color{blue}{-80} \\\Leftrightarrow & 429x& = & -600 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-600}{429} \\\Leftrightarrow & x = \frac{-200}{143} & & \\ & V = \left\{ \frac{-200}{143} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & -12x-60 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{+12x} & = & \color{red}{-12x} -60 \color{blue}{+12x} \color{blue}{+4} \\\Leftrightarrow & 27x& = & -56 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{-56}{27} \\\Leftrightarrow & x = \frac{-56}{27} & & \\ & V = \left\{ \frac{-56}{27} \right\} & \\\end{align}\)
6. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{3}{4}x+5 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 6 }{ \color{blue}{28} })& = & (\frac{21}{ \color{blue}{28} }x+\frac{140}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+6& = & 21x+140 \\\Leftrightarrow & 4x \color{red}{+6} \color{blue}{-6} \color{blue}{-21x} & = & \color{red}{21x} +140 \color{blue}{-21x} \color{blue}{-6} \\\Leftrightarrow & -17x& = & 134 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{134}{-17} \\\Leftrightarrow & x = \frac{-134}{17} & & \\ & V = \left\{ \frac{-134}{17} \right\} & \\\end{align}\)
7. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 60 }{ \color{blue}{165} })& = & (\frac{-132}{ \color{blue}{165} }x+\frac{990}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-60& = & -132x+990 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+132x} & = & \color{red}{-132x} +990 \color{blue}{+132x} \color{blue}{+60} \\\Leftrightarrow & 187x& = & 1050 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{1050}{187} \\\Leftrightarrow & x = \frac{1050}{187} & & \\ & V = \left\{ \frac{1050}{187} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{2}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 56x+588 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-56x} & = & \color{red}{56x} +588 \color{blue}{-56x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & 612 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{612}{-35} \\\Leftrightarrow & x = \frac{-612}{35} & & \\ & V = \left\{ \frac{-612}{35} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{4}x-5 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x-\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+80& = & 35x-700 \\\Leftrightarrow & 28x \color{red}{+80} \color{blue}{-80} \color{blue}{-35x} & = & \color{red}{35x} -700 \color{blue}{-35x} \color{blue}{-80} \\\Leftrightarrow & -7x& = & -780 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-780}{-7} \\\Leftrightarrow & x = \frac{780}{7} & & \\ & V = \left\{ \frac{780}{7} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{48}{ \color{blue}{36} }x+\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & 48x+36 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-48x} & = & \color{red}{48x} +36 \color{blue}{-48x} \color{blue}{+8} \\\Leftrightarrow & -39x& = & 44 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{44}{-39} \\\Leftrightarrow & x = \frac{-44}{39} & & \\ & V = \left\{ \frac{-44}{39} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x+54 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} +54 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & 50 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{50}{-3} \\\Leftrightarrow & x = \frac{-50}{3} & & \\ & V = \left\{ \frac{-50}{3} \right\} & \\\end{align}\)
12. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{7}{5}x-8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{336}{ \color{blue}{240} }x-\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & 336x-1920 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-336x} & = & \color{red}{336x} -1920 \color{blue}{-336x} \color{blue}{+75} \\\Leftrightarrow & -296x& = & -1845 \\\Leftrightarrow & \frac{-296x}{ \color{red}{-296} }& = & \frac{-1845}{-296} \\\Leftrightarrow & x = \frac{1845}{296} & & \\ & V = \left\{ \frac{1845}{296} \right\} & \\\end{align}\)