Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{1}{6}x-2 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{21}{ \color{blue}{126} }x-\frac{252}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 21x-252 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{-56} \\\Leftrightarrow & -3x& = & -308 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-308}{-3} \\\Leftrightarrow & x = \frac{308}{3} & & \\ & V = \left\{ \frac{308}{3} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{-7}{4}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & -147x+84 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{+147x} & = & \color{red}{-147x} +84 \color{blue}{+147x} \color{blue}{+36} \\\Leftrightarrow & 175x& = & 120 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{120}{175} \\\Leftrightarrow & x = \frac{24}{35} & & \\ & V = \left\{ \frac{24}{35} \right\} & \\\end{align}\)
3. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }- \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-135}{ \color{blue}{180} }x-\frac{1260}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x-40& = & -135x-1260 \\\Leftrightarrow & 36x \color{red}{-40} \color{blue}{+40} \color{blue}{+135x} & = & \color{red}{-135x} -1260 \color{blue}{+135x} \color{blue}{+40} \\\Leftrightarrow & 171x& = & -1220 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{-1220}{171} \\\Leftrightarrow & x = \frac{-1220}{171} & & \\ & V = \left\{ \frac{-1220}{171} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x-\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & -104x-260 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{+104x} & = & \color{red}{-104x} -260 \color{blue}{+104x} \color{blue}{-20} \\\Leftrightarrow & 169x& = & -280 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-280}{169} \\\Leftrightarrow & x = \frac{-280}{169} & & \\ & V = \left\{ \frac{-280}{169} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 396x-990 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-396x} & = & \color{red}{396x} -990 \color{blue}{-396x} \color{blue}{+150} \\\Leftrightarrow & -341x& = & -840 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-840}{-341} \\\Leftrightarrow & x = \frac{840}{341} & & \\ & V = \left\{ \frac{840}{341} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 112x-168 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-112x} & = & \color{red}{112x} -168 \color{blue}{-112x} \color{blue}{-48} \\\Leftrightarrow & -91x& = & -216 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-216}{-91} \\\Leftrightarrow & x = \frac{216}{91} & & \\ & V = \left\{ \frac{216}{91} \right\} & \\\end{align}\)
7. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{7}{5}x-2 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{336}{ \color{blue}{240} }x-\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & 336x-480 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-336x} & = & \color{red}{336x} -480 \color{blue}{-336x} \color{blue}{-75} \\\Leftrightarrow & -296x& = & -555 \\\Leftrightarrow & \frac{-296x}{ \color{red}{-296} }& = & \frac{-555}{-296} \\\Leftrightarrow & x = \frac{15}{8} & & \\ & V = \left\{ \frac{15}{8} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{5}{6}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & 175x+1680 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{-175x} & = & \color{red}{175x} +1680 \color{blue}{-175x} \color{blue}{+90} \\\Leftrightarrow & -133x& = & 1770 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{1770}{-133} \\\Leftrightarrow & x = \frac{-1770}{133} & & \\ & V = \left\{ \frac{-1770}{133} \right\} & \\\end{align}\)
9. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{60}{ \color{blue}{24} }x+\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 60x+72 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-60x} & = & \color{red}{60x} +72 \color{blue}{-60x} \color{blue}{+9} \\\Leftrightarrow & -52x& = & 81 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{81}{-52} \\\Leftrightarrow & x = \frac{-81}{52} & & \\ & V = \left\{ \frac{-81}{52} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{3}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 126x-420 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-126x} & = & \color{red}{126x} -420 \color{blue}{-126x} \color{blue}{-120} \\\Leftrightarrow & -91x& = & -540 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-540}{-91} \\\Leftrightarrow & x = \frac{540}{91} & & \\ & V = \left\{ \frac{540}{91} \right\} & \\\end{align}\)
11. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 70 }{ \color{blue}{126} })& = & (\frac{21}{ \color{blue}{126} }x+\frac{756}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-70& = & 21x+756 \\\Leftrightarrow & 18x \color{red}{-70} \color{blue}{+70} \color{blue}{-21x} & = & \color{red}{21x} +756 \color{blue}{-21x} \color{blue}{+70} \\\Leftrightarrow & -3x& = & 826 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{826}{-3} \\\Leftrightarrow & x = \frac{-826}{3} & & \\ & V = \left\{ \frac{-826}{3} \right\} & \\\end{align}\)
12. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 15 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x-\frac{210}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-15& = & -28x-210 \\\Leftrightarrow & 5x \color{red}{-15} \color{blue}{+15} \color{blue}{+28x} & = & \color{red}{-28x} -210 \color{blue}{+28x} \color{blue}{+15} \\\Leftrightarrow & 33x& = & -195 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-195}{33} \\\Leftrightarrow & x = \frac{-65}{11} & & \\ & V = \left\{ \frac{-65}{11} \right\} & \\\end{align}\)