Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{5}{6}x+3 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 84 }{ \color{blue}{546} })& = & (\frac{455}{ \color{blue}{546} }x+\frac{1638}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+84& = & 455x+1638 \\\Leftrightarrow & 78x \color{red}{+84} \color{blue}{-84} \color{blue}{-455x} & = & \color{red}{455x} +1638 \color{blue}{-455x} \color{blue}{-84} \\\Leftrightarrow & -377x& = & 1554 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{1554}{-377} \\\Leftrightarrow & x = \frac{-1554}{377} & & \\ & V = \left\{ \frac{-1554}{377} \right\} & \\\end{align}\)
2. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{8}& = & \frac{3}{4}x+5 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }- \frac{ 15 }{ \color{blue}{40} })& = & (\frac{30}{ \color{blue}{40} }x+\frac{200}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x-15& = & 30x+200 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-30x} & = & \color{red}{30x} +200 \color{blue}{-30x} \color{blue}{+15} \\\Leftrightarrow & -22x& = & 215 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{215}{-22} \\\Leftrightarrow & x = \frac{-215}{22} & & \\ & V = \left\{ \frac{-215}{22} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+12& = & -35x-210 \\\Leftrightarrow & 6x \color{red}{+12} \color{blue}{-12} \color{blue}{+35x} & = & \color{red}{-35x} -210 \color{blue}{+35x} \color{blue}{-12} \\\Leftrightarrow & 41x& = & -222 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{-222}{41} \\\Leftrightarrow & x = \frac{-222}{41} & & \\ & V = \left\{ \frac{-222}{41} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{3}{4}x-2 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }+ \frac{ 60 }{ \color{blue}{220} })& = & (\frac{165}{ \color{blue}{220} }x-\frac{440}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x+60& = & 165x-440 \\\Leftrightarrow & 44x \color{red}{+60} \color{blue}{-60} \color{blue}{-165x} & = & \color{red}{165x} -440 \color{blue}{-165x} \color{blue}{-60} \\\Leftrightarrow & -121x& = & -500 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-500}{-121} \\\Leftrightarrow & x = \frac{500}{121} & & \\ & V = \left\{ \frac{500}{121} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{7}{3}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{308}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & 308x+792 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{-308x} & = & \color{red}{308x} +792 \color{blue}{-308x} \color{blue}{-48} \\\Leftrightarrow & -275x& = & 744 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{744}{-275} \\\Leftrightarrow & x = \frac{-744}{275} & & \\ & V = \left\{ \frac{-744}{275} \right\} & \\\end{align}\)
6. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{5}{4}x+1 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 84 }{ \color{blue}{364} })& = & (\frac{455}{ \color{blue}{364} }x+\frac{364}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-84& = & 455x+364 \\\Leftrightarrow & 52x \color{red}{-84} \color{blue}{+84} \color{blue}{-455x} & = & \color{red}{455x} +364 \color{blue}{-455x} \color{blue}{+84} \\\Leftrightarrow & -403x& = & 448 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{448}{-403} \\\Leftrightarrow & x = \frac{-448}{403} & & \\ & V = \left\{ \frac{-448}{403} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{5}{2}x-3 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{165}{ \color{blue}{66} }x-\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-12& = & 165x-198 \\\Leftrightarrow & 22x \color{red}{-12} \color{blue}{+12} \color{blue}{-165x} & = & \color{red}{165x} -198 \color{blue}{-165x} \color{blue}{+12} \\\Leftrightarrow & -143x& = & -186 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-186}{-143} \\\Leftrightarrow & x = \frac{186}{143} & & \\ & V = \left\{ \frac{186}{143} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-273}{ \color{blue}{156} }x-\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+24& = & -273x-468 \\\Leftrightarrow & 52x \color{red}{+24} \color{blue}{-24} \color{blue}{+273x} & = & \color{red}{-273x} -468 \color{blue}{+273x} \color{blue}{-24} \\\Leftrightarrow & 325x& = & -492 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{-492}{325} \\\Leftrightarrow & x = \frac{-492}{325} & & \\ & V = \left\{ \frac{-492}{325} \right\} & \\\end{align}\)
9. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-9}{ \color{blue}{12} }x-\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & -9x-60 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{+9x} & = & \color{red}{-9x} -60 \color{blue}{+9x} \color{blue}{-5} \\\Leftrightarrow & 13x& = & -65 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-65}{13} \\\Leftrightarrow & x = -5 & & \\ & V = \left\{ -5 \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -126x-630 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+126x} & = & \color{red}{-126x} -630 \color{blue}{+126x} \color{blue}{-40} \\\Leftrightarrow & 141x& = & -670 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-670}{141} \\\Leftrightarrow & x = \frac{-670}{141} & & \\ & V = \left\{ \frac{-670}{141} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 15x-150 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-15x} & = & \color{red}{15x} -150 \color{blue}{-15x} \color{blue}{-25} \\\Leftrightarrow & -9x& = & -175 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-175}{-9} \\\Leftrightarrow & x = \frac{175}{9} & & \\ & V = \left\{ \frac{175}{9} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 24x-48 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-24x} & = & \color{red}{24x} -48 \color{blue}{-24x} \color{blue}{+9} \\\Leftrightarrow & -8x& = & -39 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-39}{-8} \\\Leftrightarrow & x = \frac{39}{8} & & \\ & V = \left\{ \frac{39}{8} \right\} & \\\end{align}\)