Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x+144 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} +144 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & 135 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{135}{-8} \\\Leftrightarrow & x = \frac{-135}{8} & & \\ & V = \left\{ \frac{-135}{8} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 7x-98 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-7x} & = & \color{red}{7x} -98 \color{blue}{-7x} \color{blue}{-4} \\\Leftrightarrow & -5x& = & -102 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-102}{-5} \\\Leftrightarrow & x = \frac{102}{5} & & \\ & V = \left\{ \frac{102}{5} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & -84x-1470 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{+84x} & = & \color{red}{-84x} -1470 \color{blue}{+84x} \color{blue}{+120} \\\Leftrightarrow & 119x& = & -1350 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-1350}{119} \\\Leftrightarrow & x = \frac{-1350}{119} & & \\ & V = \left\{ \frac{-1350}{119} \right\} & \\\end{align}\)
4. \(\text{420 is het kleinste gemene veelvoud van 7, 12 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{12}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 175 }{ \color{blue}{420} })& = & (\frac{-168}{ \color{blue}{420} }x+\frac{3360}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+175& = & -168x+3360 \\\Leftrightarrow & 60x \color{red}{+175} \color{blue}{-175} \color{blue}{+168x} & = & \color{red}{-168x} +3360 \color{blue}{+168x} \color{blue}{-175} \\\Leftrightarrow & 228x& = & 3185 \\\Leftrightarrow & \frac{228x}{ \color{red}{228} }& = & \frac{3185}{228} \\\Leftrightarrow & x = \frac{3185}{228} & & \\ & V = \left\{ \frac{3185}{228} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{-3}{4}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-16& = & -45x+360 \\\Leftrightarrow & 20x \color{red}{-16} \color{blue}{+16} \color{blue}{+45x} & = & \color{red}{-45x} +360 \color{blue}{+45x} \color{blue}{+16} \\\Leftrightarrow & 65x& = & 376 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{376}{65} \\\Leftrightarrow & x = \frac{376}{65} & & \\ & V = \left\{ \frac{376}{65} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{5}{6}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{325}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-90& = & 325x+2340 \\\Leftrightarrow & 78x \color{red}{-90} \color{blue}{+90} \color{blue}{-325x} & = & \color{red}{325x} +2340 \color{blue}{-325x} \color{blue}{+90} \\\Leftrightarrow & -247x& = & 2430 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{2430}{-247} \\\Leftrightarrow & x = \frac{-2430}{247} & & \\ & V = \left\{ \frac{-2430}{247} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & 18x-90 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{-18x} & = & \color{red}{18x} -90 \color{blue}{-18x} \color{blue}{+20} \\\Leftrightarrow & 27x& = & -70 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{-70}{27} \\\Leftrightarrow & x = \frac{-70}{27} & & \\ & V = \left\{ \frac{-70}{27} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & -56x-588 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{+56x} & = & \color{red}{-56x} -588 \color{blue}{+56x} \color{blue}{-36} \\\Leftrightarrow & 77x& = & -624 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-624}{77} \\\Leftrightarrow & x = \frac{-624}{77} & & \\ & V = \left\{ \frac{-624}{77} \right\} & \\\end{align}\)
9. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{200}{ \color{blue}{80} }x-\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & 200x-560 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-200x} & = & \color{red}{200x} -560 \color{blue}{-200x} \color{blue}{+15} \\\Leftrightarrow & -184x& = & -545 \\\Leftrightarrow & \frac{-184x}{ \color{red}{-184} }& = & \frac{-545}{-184} \\\Leftrightarrow & x = \frac{545}{184} & & \\ & V = \left\{ \frac{545}{184} \right\} & \\\end{align}\)
10. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{1}{6}x-5 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 168 }{ \color{blue}{546} })& = & (\frac{91}{ \color{blue}{546} }x-\frac{2730}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+168& = & 91x-2730 \\\Leftrightarrow & 78x \color{red}{+168} \color{blue}{-168} \color{blue}{-91x} & = & \color{red}{91x} -2730 \color{blue}{-91x} \color{blue}{-168} \\\Leftrightarrow & -13x& = & -2898 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-2898}{-13} \\\Leftrightarrow & x = \frac{2898}{13} & & \\ & V = \left\{ \frac{2898}{13} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & 252x-840 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{-252x} & = & \color{red}{252x} -840 \color{blue}{-252x} \color{blue}{+45} \\\Leftrightarrow & -217x& = & -795 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-795}{-217} \\\Leftrightarrow & x = \frac{795}{217} & & \\ & V = \left\{ \frac{795}{217} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & -28x-294 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{+28x} & = & \color{red}{-28x} -294 \color{blue}{+28x} \color{blue}{+9} \\\Leftrightarrow & 49x& = & -285 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-285}{49} \\\Leftrightarrow & x = \frac{-285}{49} & & \\ & V = \left\{ \frac{-285}{49} \right\} & \\\end{align}\)