Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & 252x+1260 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{-252x} & = & \color{red}{252x} +1260 \color{blue}{-252x} \color{blue}{+150} \\\Leftrightarrow & -217x& = & 1410 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{1410}{-217} \\\Leftrightarrow & x = \frac{-1410}{217} & & \\ & V = \left\{ \frac{-1410}{217} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x-\frac{1040}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-30& = & -52x-1040 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{+52x} & = & \color{red}{-52x} -1040 \color{blue}{+52x} \color{blue}{+30} \\\Leftrightarrow & 117x& = & -1010 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-1010}{117} \\\Leftrightarrow & x = \frac{-1010}{117} & & \\ & V = \left\{ \frac{-1010}{117} \right\} & \\\end{align}\)
3. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 56 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{308}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-56& = & 77x-308 \\\Leftrightarrow & 22x \color{red}{-56} \color{blue}{+56} \color{blue}{-77x} & = & \color{red}{77x} -308 \color{blue}{-77x} \color{blue}{+56} \\\Leftrightarrow & -55x& = & -252 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-252}{-55} \\\Leftrightarrow & x = \frac{252}{55} & & \\ & V = \left\{ \frac{252}{55} \right\} & \\\end{align}\)
4. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{7}{2}x+3 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x+\frac{42}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 49x+42 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-49x} & = & \color{red}{49x} +42 \color{blue}{-49x} \color{blue}{+4} \\\Leftrightarrow & -47x& = & 46 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{46}{-47} \\\Leftrightarrow & x = \frac{-46}{47} & & \\ & V = \left\{ \frac{-46}{47} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-84 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -84 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -108 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-108}{-7} \\\Leftrightarrow & x = \frac{108}{7} & & \\ & V = \left\{ \frac{108}{7} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 9x-36 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-9x} & = & \color{red}{9x} -36 \color{blue}{-9x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & -28 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-28}{-3} \\\Leftrightarrow & x = \frac{28}{3} & & \\ & V = \left\{ \frac{28}{3} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-105}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+8& = & -105x+420 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{+105x} & = & \color{red}{-105x} +420 \color{blue}{+105x} \color{blue}{-8} \\\Leftrightarrow & 117x& = & 412 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{412}{117} \\\Leftrightarrow & x = \frac{412}{117} & & \\ & V = \left\{ \frac{412}{117} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+60& = & 168x-420 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-168x} & = & \color{red}{168x} -420 \color{blue}{-168x} \color{blue}{-60} \\\Leftrightarrow & -133x& = & -480 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-480}{-133} \\\Leftrightarrow & x = \frac{480}{133} & & \\ & V = \left\{ \frac{480}{133} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{8}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{144}{ \color{blue}{120} }x+\frac{240}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-45& = & 144x+240 \\\Leftrightarrow & 20x \color{red}{-45} \color{blue}{+45} \color{blue}{-144x} & = & \color{red}{144x} +240 \color{blue}{-144x} \color{blue}{+45} \\\Leftrightarrow & -124x& = & 285 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{285}{-124} \\\Leftrightarrow & x = \frac{-285}{124} & & \\ & V = \left\{ \frac{-285}{124} \right\} & \\\end{align}\)
10. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-60& = & -88x-528 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{+88x} & = & \color{red}{-88x} -528 \color{blue}{+88x} \color{blue}{+60} \\\Leftrightarrow & 121x& = & -468 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-468}{121} \\\Leftrightarrow & x = \frac{-468}{121} & & \\ & V = \left\{ \frac{-468}{121} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 14x+252 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-14x} & = & \color{red}{14x} +252 \color{blue}{-14x} \color{blue}{-24} \\\Leftrightarrow & 7x& = & 228 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{228}{7} \\\Leftrightarrow & x = \frac{228}{7} & & \\ & V = \left\{ \frac{228}{7} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{4}{5}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{56}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & 56x+490 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{-56x} & = & \color{red}{56x} +490 \color{blue}{-56x} \color{blue}{+15} \\\Leftrightarrow & -21x& = & 505 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{505}{-21} \\\Leftrightarrow & x = \frac{-505}{21} & & \\ & V = \left\{ \frac{-505}{21} \right\} & \\\end{align}\)