Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{1}{6}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+18& = & 7x+294 \\\Leftrightarrow & 6x \color{red}{+18} \color{blue}{-18} \color{blue}{-7x} & = & \color{red}{7x} +294 \color{blue}{-7x} \color{blue}{-18} \\\Leftrightarrow & -x& = & 276 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{276}{-1} \\\Leftrightarrow & x = -276 & & \\ & V = \left\{ -276 \right\} & \\\end{align}\)
2. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 100 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+100& = & 168x+420 \\\Leftrightarrow & 35x \color{red}{+100} \color{blue}{-100} \color{blue}{-168x} & = & \color{red}{168x} +420 \color{blue}{-168x} \color{blue}{-100} \\\Leftrightarrow & -133x& = & 320 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{320}{-133} \\\Leftrightarrow & x = \frac{-320}{133} & & \\ & V = \left\{ \frac{-320}{133} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -28x-350 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+28x} & = & \color{red}{-28x} -350 \color{blue}{+28x} \color{blue}{+20} \\\Leftrightarrow & 63x& = & -330 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{-330}{63} \\\Leftrightarrow & x = \frac{-110}{21} & & \\ & V = \left\{ \frac{-110}{21} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{7}{2}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{294}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & 294x+336 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{-294x} & = & \color{red}{294x} +336 \color{blue}{-294x} \color{blue}{+35} \\\Leftrightarrow & -282x& = & 371 \\\Leftrightarrow & \frac{-282x}{ \color{red}{-282} }& = & \frac{371}{-282} \\\Leftrightarrow & x = \frac{-371}{282} & & \\ & V = \left\{ \frac{-371}{282} \right\} & \\\end{align}\)
5. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }+ \frac{ 14 }{ \color{blue}{63} })& = & (\frac{147}{ \color{blue}{63} }x-\frac{504}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x+14& = & 147x-504 \\\Leftrightarrow & 9x \color{red}{+14} \color{blue}{-14} \color{blue}{-147x} & = & \color{red}{147x} -504 \color{blue}{-147x} \color{blue}{-14} \\\Leftrightarrow & -138x& = & -518 \\\Leftrightarrow & \frac{-138x}{ \color{red}{-138} }& = & \frac{-518}{-138} \\\Leftrightarrow & x = \frac{259}{69} & & \\ & V = \left\{ \frac{259}{69} \right\} & \\\end{align}\)
6. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{5}{2}x+2 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{60}{ \color{blue}{24} }x+\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 60x+48 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-60x} & = & \color{red}{60x} +48 \color{blue}{-60x} \color{blue}{+9} \\\Leftrightarrow & -52x& = & 57 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{57}{-52} \\\Leftrightarrow & x = \frac{-57}{52} & & \\ & V = \left\{ \frac{-57}{52} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & -147x-252 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{+147x} & = & \color{red}{-147x} -252 \color{blue}{+147x} \color{blue}{-48} \\\Leftrightarrow & 175x& = & -300 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-300}{175} \\\Leftrightarrow & x = \frac{-12}{7} & & \\ & V = \left\{ \frac{-12}{7} \right\} & \\\end{align}\)
8. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{-175}{ \color{blue}{105} }x-\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+30& = & -175x-525 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{+175x} & = & \color{red}{-175x} -525 \color{blue}{+175x} \color{blue}{-30} \\\Leftrightarrow & 196x& = & -555 \\\Leftrightarrow & \frac{196x}{ \color{red}{196} }& = & \frac{-555}{196} \\\Leftrightarrow & x = \frac{-555}{196} & & \\ & V = \left\{ \frac{-555}{196} \right\} & \\\end{align}\)
9. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 6 }{ \color{blue}{21} })& = & (\frac{28}{ \color{blue}{21} }x+\frac{168}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+6& = & 28x+168 \\\Leftrightarrow & 3x \color{red}{+6} \color{blue}{-6} \color{blue}{-28x} & = & \color{red}{28x} +168 \color{blue}{-28x} \color{blue}{-6} \\\Leftrightarrow & -25x& = & 162 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{162}{-25} \\\Leftrightarrow & x = \frac{-162}{25} & & \\ & V = \left\{ \frac{-162}{25} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{3}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 126x-210 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-126x} & = & \color{red}{126x} -210 \color{blue}{-126x} \color{blue}{+60} \\\Leftrightarrow & -91x& = & -150 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-150}{-91} \\\Leftrightarrow & x = \frac{150}{91} & & \\ & V = \left\{ \frac{150}{91} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{13}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x-\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-150& = & -156x-390 \\\Leftrightarrow & 65x \color{red}{-150} \color{blue}{+150} \color{blue}{+156x} & = & \color{red}{-156x} -390 \color{blue}{+156x} \color{blue}{+150} \\\Leftrightarrow & 221x& = & -240 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-240}{221} \\\Leftrightarrow & x = \frac{-240}{221} & & \\ & V = \left\{ \frac{-240}{221} \right\} & \\\end{align}\)
12. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{728}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 91x-728 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-91x} & = & \color{red}{91x} -728 \color{blue}{-91x} \color{blue}{+42} \\\Leftrightarrow & -65x& = & -686 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-686}{-65} \\\Leftrightarrow & x = \frac{686}{65} & & \\ & V = \left\{ \frac{686}{65} \right\} & \\\end{align}\)