Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & -50x+120 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{+50x} & = & \color{red}{-50x} +120 \color{blue}{+50x} \color{blue}{+4} \\\Leftrightarrow & 65x& = & 124 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{124}{65} \\\Leftrightarrow & x = \frac{124}{65} & & \\ & V = \left\{ \frac{124}{65} \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-8}{3}x-4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-48}{ \color{blue}{18} }x-\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & -48x-72 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+48x} & = & \color{red}{-48x} -72 \color{blue}{+48x} \color{blue}{+8} \\\Leftrightarrow & 57x& = & -64 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-64}{57} \\\Leftrightarrow & x = \frac{-64}{57} & & \\ & V = \left\{ \frac{-64}{57} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{14}& = & \frac{4}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-75& = & 168x+1680 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{-168x} & = & \color{red}{168x} +1680 \color{blue}{-168x} \color{blue}{+75} \\\Leftrightarrow & -133x& = & 1755 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{1755}{-133} \\\Leftrightarrow & x = \frac{-1755}{133} & & \\ & V = \left\{ \frac{-1755}{133} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 20x+240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-20x} & = & \color{red}{20x} +240 \color{blue}{-20x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & 248 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{248}{-5} \\\Leftrightarrow & x = \frac{-248}{5} & & \\ & V = \left\{ \frac{-248}{5} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 35x+1260 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-35x} & = & \color{red}{35x} +1260 \color{blue}{-35x} \color{blue}{-60} \\\Leftrightarrow & 7x& = & 1200 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{1200}{7} \\\Leftrightarrow & x = \frac{1200}{7} & & \\ & V = \left\{ \frac{1200}{7} \right\} & \\\end{align}\)
6. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 70 }{ \color{blue}{126} })& = & (\frac{-105}{ \color{blue}{126} }x+\frac{882}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-70& = & -105x+882 \\\Leftrightarrow & 18x \color{red}{-70} \color{blue}{+70} \color{blue}{+105x} & = & \color{red}{-105x} +882 \color{blue}{+105x} \color{blue}{+70} \\\Leftrightarrow & 123x& = & 952 \\\Leftrightarrow & \frac{123x}{ \color{red}{123} }& = & \frac{952}{123} \\\Leftrightarrow & x = \frac{952}{123} & & \\ & V = \left\{ \frac{952}{123} \right\} & \\\end{align}\)
7. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-40& = & 65x-910 \\\Leftrightarrow & 26x \color{red}{-40} \color{blue}{+40} \color{blue}{-65x} & = & \color{red}{65x} -910 \color{blue}{-65x} \color{blue}{+40} \\\Leftrightarrow & -39x& = & -870 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-870}{-39} \\\Leftrightarrow & x = \frac{290}{13} & & \\ & V = \left\{ \frac{290}{13} \right\} & \\\end{align}\)
8. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-8}{ \color{blue}{12} }x-\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-5& = & -8x-84 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{+8x} & = & \color{red}{-8x} -84 \color{blue}{+8x} \color{blue}{+5} \\\Leftrightarrow & 11x& = & -79 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-79}{11} \\\Leftrightarrow & x = \frac{-79}{11} & & \\ & V = \left\{ \frac{-79}{11} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{7}{2}x+6 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{273}{ \color{blue}{78} }x+\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+24& = & 273x+468 \\\Leftrightarrow & 26x \color{red}{+24} \color{blue}{-24} \color{blue}{-273x} & = & \color{red}{273x} +468 \color{blue}{-273x} \color{blue}{-24} \\\Leftrightarrow & -247x& = & 444 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{444}{-247} \\\Leftrightarrow & x = \frac{-444}{247} & & \\ & V = \left\{ \frac{-444}{247} \right\} & \\\end{align}\)
10. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 63 }{ \color{blue}{273} })& = & (\frac{-455}{ \color{blue}{273} }x-\frac{1365}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-63& = & -455x-1365 \\\Leftrightarrow & 39x \color{red}{-63} \color{blue}{+63} \color{blue}{+455x} & = & \color{red}{-455x} -1365 \color{blue}{+455x} \color{blue}{+63} \\\Leftrightarrow & 494x& = & -1302 \\\Leftrightarrow & \frac{494x}{ \color{red}{494} }& = & \frac{-1302}{494} \\\Leftrightarrow & x = \frac{-651}{247} & & \\ & V = \left\{ \frac{-651}{247} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+24& = & 7x-336 \\\Leftrightarrow & 6x \color{red}{+24} \color{blue}{-24} \color{blue}{-7x} & = & \color{red}{7x} -336 \color{blue}{-7x} \color{blue}{-24} \\\Leftrightarrow & -x& = & -360 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-360}{-1} \\\Leftrightarrow & x = 360 & & \\ & V = \left\{ 360 \right\} & \\\end{align}\)
12. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{-3}{4}x-8 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }- \frac{ 80 }{ \color{blue}{220} })& = & (\frac{-165}{ \color{blue}{220} }x-\frac{1760}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x-80& = & -165x-1760 \\\Leftrightarrow & 44x \color{red}{-80} \color{blue}{+80} \color{blue}{+165x} & = & \color{red}{-165x} -1760 \color{blue}{+165x} \color{blue}{+80} \\\Leftrightarrow & 209x& = & -1680 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-1680}{209} \\\Leftrightarrow & x = \frac{-1680}{209} & & \\ & V = \left\{ \frac{-1680}{209} \right\} & \\\end{align}\)