Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-2}{5}x-4 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x-\frac{520}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-20& = & -52x-520 \\\Leftrightarrow & 65x \color{red}{-20} \color{blue}{+20} \color{blue}{+52x} & = & \color{red}{-52x} -520 \color{blue}{+52x} \color{blue}{+20} \\\Leftrightarrow & 117x& = & -500 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-500}{117} \\\Leftrightarrow & x = \frac{-500}{117} & & \\ & V = \left\{ \frac{-500}{117} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{2}{5}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & 12x-30 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{-12x} & = & \color{red}{12x} -30 \color{blue}{-12x} \color{blue}{-8} \\\Leftrightarrow & -7x& = & -38 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-38}{-7} \\\Leftrightarrow & x = \frac{38}{7} & & \\ & V = \left\{ \frac{38}{7} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-56}{ \color{blue}{140} }x+\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-60& = & -56x+1120 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+56x} & = & \color{red}{-56x} +1120 \color{blue}{+56x} \color{blue}{+60} \\\Leftrightarrow & 91x& = & 1180 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{1180}{91} \\\Leftrightarrow & x = \frac{1180}{91} & & \\ & V = \left\{ \frac{1180}{91} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 63 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+63& = & -175x+1260 \\\Leftrightarrow & 30x \color{red}{+63} \color{blue}{-63} \color{blue}{+175x} & = & \color{red}{-175x} +1260 \color{blue}{+175x} \color{blue}{-63} \\\Leftrightarrow & 205x& = & 1197 \\\Leftrightarrow & \frac{205x}{ \color{red}{205} }& = & \frac{1197}{205} \\\Leftrightarrow & x = \frac{1197}{205} & & \\ & V = \left\{ \frac{1197}{205} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & -175x-1050 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{+175x} & = & \color{red}{-175x} -1050 \color{blue}{+175x} \color{blue}{-90} \\\Leftrightarrow & 217x& = & -1140 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-1140}{217} \\\Leftrightarrow & x = \frac{-1140}{217} & & \\ & V = \left\{ \frac{-1140}{217} \right\} & \\\end{align}\)
6. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 80 }{ \color{blue}{220} })& = & (\frac{264}{ \color{blue}{220} }x+\frac{440}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-80& = & 264x+440 \\\Leftrightarrow & 55x \color{red}{-80} \color{blue}{+80} \color{blue}{-264x} & = & \color{red}{264x} +440 \color{blue}{-264x} \color{blue}{+80} \\\Leftrightarrow & -209x& = & 520 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{520}{-209} \\\Leftrightarrow & x = \frac{-520}{209} & & \\ & V = \left\{ \frac{-520}{209} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{15}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+20& = & 15x+270 \\\Leftrightarrow & 18x \color{red}{+20} \color{blue}{-20} \color{blue}{-15x} & = & \color{red}{15x} +270 \color{blue}{-15x} \color{blue}{-20} \\\Leftrightarrow & 3x& = & 250 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{250}{3} \\\Leftrightarrow & x = \frac{250}{3} & & \\ & V = \left\{ \frac{250}{3} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{104}{ \color{blue}{78} }x+\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & 104x+546 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{-104x} & = & \color{red}{104x} +546 \color{blue}{-104x} \color{blue}{+18} \\\Leftrightarrow & -65x& = & 564 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{564}{-65} \\\Leftrightarrow & x = \frac{-564}{65} & & \\ & V = \left\{ \frac{-564}{65} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{64}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 64x-240 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-64x} & = & \color{red}{64x} -240 \color{blue}{-64x} \color{blue}{-9} \\\Leftrightarrow & -40x& = & -249 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-249}{-40} \\\Leftrightarrow & x = \frac{249}{40} & & \\ & V = \left\{ \frac{249}{40} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 21x-252 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{-18} \\\Leftrightarrow & -7x& = & -270 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-270}{-7} \\\Leftrightarrow & x = \frac{270}{7} & & \\ & V = \left\{ \frac{270}{7} \right\} & \\\end{align}\)
11. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{5}{2}x+8 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{325}{ \color{blue}{130} }x+\frac{1040}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-20& = & 325x+1040 \\\Leftrightarrow & 26x \color{red}{-20} \color{blue}{+20} \color{blue}{-325x} & = & \color{red}{325x} +1040 \color{blue}{-325x} \color{blue}{+20} \\\Leftrightarrow & -299x& = & 1060 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{1060}{-299} \\\Leftrightarrow & x = \frac{-1060}{299} & & \\ & V = \left\{ \frac{-1060}{299} \right\} & \\\end{align}\)
12. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{4}{3}x-7 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{16}{ \color{blue}{12} }x-\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & 16x-84 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{-16x} & = & \color{red}{16x} -84 \color{blue}{-16x} \color{blue}{+10} \\\Leftrightarrow & -13x& = & -74 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-74}{-13} \\\Leftrightarrow & x = \frac{74}{13} & & \\ & V = \left\{ \frac{74}{13} \right\} & \\\end{align}\)