Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -140x-168 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+140x} & = & \color{red}{-140x} -168 \color{blue}{+140x} \color{blue}{+24} \\\Leftrightarrow & 161x& = & -144 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-144}{161} \\\Leftrightarrow & x = \frac{-144}{161} & & \\ & V = \left\{ \frac{-144}{161} \right\} & \\\end{align}\)
2. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{21}{ \color{blue}{105} }x+\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+30& = & 21x+525 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-21x} & = & \color{red}{21x} +525 \color{blue}{-21x} \color{blue}{-30} \\\Leftrightarrow & 14x& = & 495 \\\Leftrightarrow & \frac{14x}{ \color{red}{14} }& = & \frac{495}{14} \\\Leftrightarrow & x = \frac{495}{14} & & \\ & V = \left\{ \frac{495}{14} \right\} & \\\end{align}\)
3. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{20}{ \color{blue}{40} }x+\frac{240}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+25& = & 20x+240 \\\Leftrightarrow & 8x \color{red}{+25} \color{blue}{-25} \color{blue}{-20x} & = & \color{red}{20x} +240 \color{blue}{-20x} \color{blue}{-25} \\\Leftrightarrow & -12x& = & 215 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{215}{-12} \\\Leftrightarrow & x = \frac{-215}{12} & & \\ & V = \left\{ \frac{-215}{12} \right\} & \\\end{align}\)
4. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{8}& = & \frac{-5}{6}x-4 \\\Leftrightarrow & \color{blue}{168.} (\frac{24x}{ \color{blue}{168} }+ \frac{ 105 }{ \color{blue}{168} })& = & (\frac{-140}{ \color{blue}{168} }x-\frac{672}{ \color{blue}{168} }) \color{blue}{.168} \\\Leftrightarrow & 24x+105& = & -140x-672 \\\Leftrightarrow & 24x \color{red}{+105} \color{blue}{-105} \color{blue}{+140x} & = & \color{red}{-140x} -672 \color{blue}{+140x} \color{blue}{-105} \\\Leftrightarrow & 164x& = & -777 \\\Leftrightarrow & \frac{164x}{ \color{red}{164} }& = & \frac{-777}{164} \\\Leftrightarrow & x = \frac{-777}{164} & & \\ & V = \left\{ \frac{-777}{164} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{14}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 25 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-25& = & 14x+420 \\\Leftrightarrow & 35x \color{red}{-25} \color{blue}{+25} \color{blue}{-14x} & = & \color{red}{14x} +420 \color{blue}{-14x} \color{blue}{+25} \\\Leftrightarrow & 21x& = & 445 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{445}{21} \\\Leftrightarrow & x = \frac{445}{21} & & \\ & V = \left\{ \frac{445}{21} \right\} & \\\end{align}\)
6. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{-7}{4}x-1 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-140}{ \color{blue}{80} }x-\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & -140x-80 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{+140x} & = & \color{red}{-140x} -80 \color{blue}{+140x} \color{blue}{-25} \\\Leftrightarrow & 156x& = & -105 \\\Leftrightarrow & \frac{156x}{ \color{red}{156} }& = & \frac{-105}{156} \\\Leftrightarrow & x = \frac{-35}{52} & & \\ & V = \left\{ \frac{-35}{52} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -28x+42 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+28x} & = & \color{red}{-28x} +42 \color{blue}{+28x} \color{blue}{-18} \\\Leftrightarrow & 49x& = & 24 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{24}{49} \\\Leftrightarrow & x = \frac{24}{49} & & \\ & V = \left\{ \frac{24}{49} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & -24x+180 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{+24x} & = & \color{red}{-24x} +180 \color{blue}{+24x} \color{blue}{-8} \\\Leftrightarrow & 29x& = & 172 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{172}{29} \\\Leftrightarrow & x = \frac{172}{29} & & \\ & V = \left\{ \frac{172}{29} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{7}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 147x-252 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-147x} & = & \color{red}{147x} -252 \color{blue}{-147x} \color{blue}{-18} \\\Leftrightarrow & -133x& = & -270 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-270}{-133} \\\Leftrightarrow & x = \frac{270}{133} & & \\ & V = \left\{ \frac{270}{133} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 196x+252 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-196x} & = & \color{red}{196x} +252 \color{blue}{-196x} \color{blue}{+48} \\\Leftrightarrow & -175x& = & 300 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{300}{-175} \\\Leftrightarrow & x = \frac{-12}{7} & & \\ & V = \left\{ \frac{-12}{7} \right\} & \\\end{align}\)
11. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{8}& = & \frac{3}{4}x-4 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{18}{ \color{blue}{24} }x-\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+15& = & 18x-96 \\\Leftrightarrow & 8x \color{red}{+15} \color{blue}{-15} \color{blue}{-18x} & = & \color{red}{18x} -96 \color{blue}{-18x} \color{blue}{-15} \\\Leftrightarrow & -10x& = & -111 \\\Leftrightarrow & \frac{-10x}{ \color{red}{-10} }& = & \frac{-111}{-10} \\\Leftrightarrow & x = \frac{111}{10} & & \\ & V = \left\{ \frac{111}{10} \right\} & \\\end{align}\)
12. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x-\frac{48}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x-5& = & 3x-48 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-3x} & = & \color{red}{3x} -48 \color{blue}{-3x} \color{blue}{+5} \\\Leftrightarrow & -x& = & -43 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-43}{-1} \\\Leftrightarrow & x = 43 & & \\ & V = \left\{ 43 \right\} & \\\end{align}\)