Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{2}{3}x-8 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{88}{ \color{blue}{132} }x-\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & 88x-1056 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{-88x} & = & \color{red}{88x} -1056 \color{blue}{-88x} \color{blue}{-48} \\\Leftrightarrow & -55x& = & -1104 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-1104}{-55} \\\Leftrightarrow & x = \frac{1104}{55} & & \\ & V = \left\{ \frac{1104}{55} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & 26x-468 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-26x} & = & \color{red}{26x} -468 \color{blue}{-26x} \color{blue}{-24} \\\Leftrightarrow & 13x& = & -492 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-492}{13} \\\Leftrightarrow & x = \frac{-492}{13} & & \\ & V = \left\{ \frac{-492}{13} \right\} & \\\end{align}\)
3. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{11}& = & \frac{4}{5}x-2 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 60 }{ \color{blue}{220} })& = & (\frac{176}{ \color{blue}{220} }x-\frac{440}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+60& = & 176x-440 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-176x} & = & \color{red}{176x} -440 \color{blue}{-176x} \color{blue}{-60} \\\Leftrightarrow & -121x& = & -500 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-500}{-121} \\\Leftrightarrow & x = \frac{500}{121} & & \\ & V = \left\{ \frac{500}{121} \right\} & \\\end{align}\)
4. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 63 }{ \color{blue}{336} })& = & (\frac{-896}{ \color{blue}{336} }x+\frac{672}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-63& = & -896x+672 \\\Leftrightarrow & 48x \color{red}{-63} \color{blue}{+63} \color{blue}{+896x} & = & \color{red}{-896x} +672 \color{blue}{+896x} \color{blue}{+63} \\\Leftrightarrow & 944x& = & 735 \\\Leftrightarrow & \frac{944x}{ \color{red}{944} }& = & \frac{735}{944} \\\Leftrightarrow & x = \frac{735}{944} & & \\ & V = \left\{ \frac{735}{944} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & 35x-630 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{-35x} & = & \color{red}{35x} -630 \color{blue}{-35x} \color{blue}{-120} \\\Leftrightarrow & 7x& = & -750 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-750}{7} \\\Leftrightarrow & x = \frac{-750}{7} & & \\ & V = \left\{ \frac{-750}{7} \right\} & \\\end{align}\)
6. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{-7}{4}x+8 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{-455}{ \color{blue}{260} }x+\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-60& = & -455x+2080 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{+455x} & = & \color{red}{-455x} +2080 \color{blue}{+455x} \color{blue}{+60} \\\Leftrightarrow & 507x& = & 2140 \\\Leftrightarrow & \frac{507x}{ \color{red}{507} }& = & \frac{2140}{507} \\\Leftrightarrow & x = \frac{2140}{507} & & \\ & V = \left\{ \frac{2140}{507} \right\} & \\\end{align}\)
7. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 12 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x-\frac{56}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+12& = & -21x-56 \\\Leftrightarrow & 4x \color{red}{+12} \color{blue}{-12} \color{blue}{+21x} & = & \color{red}{-21x} -56 \color{blue}{+21x} \color{blue}{-12} \\\Leftrightarrow & 25x& = & -68 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{-68}{25} \\\Leftrightarrow & x = \frac{-68}{25} & & \\ & V = \left\{ \frac{-68}{25} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 14x-294 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-14x} & = & \color{red}{14x} -294 \color{blue}{-14x} \color{blue}{-24} \\\Leftrightarrow & 7x& = & -318 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-318}{7} \\\Leftrightarrow & x = \frac{-318}{7} & & \\ & V = \left\{ \frac{-318}{7} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x-84 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} -84 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & -96 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-96}{-7} \\\Leftrightarrow & x = \frac{96}{7} & & \\ & V = \left\{ \frac{96}{7} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{7}& = & \frac{-5}{6}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-150& = & -175x-420 \\\Leftrightarrow & 42x \color{red}{-150} \color{blue}{+150} \color{blue}{+175x} & = & \color{red}{-175x} -420 \color{blue}{+175x} \color{blue}{+150} \\\Leftrightarrow & 217x& = & -270 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-270}{217} \\\Leftrightarrow & x = \frac{-270}{217} & & \\ & V = \left\{ \frac{-270}{217} \right\} & \\\end{align}\)
11. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{100}{ \color{blue}{120} }x+\frac{240}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x-75& = & 100x+240 \\\Leftrightarrow & 24x \color{red}{-75} \color{blue}{+75} \color{blue}{-100x} & = & \color{red}{100x} +240 \color{blue}{-100x} \color{blue}{+75} \\\Leftrightarrow & -76x& = & 315 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{315}{-76} \\\Leftrightarrow & x = \frac{-315}{76} & & \\ & V = \left\{ \frac{-315}{76} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{9}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-50& = & -75x+270 \\\Leftrightarrow & 18x \color{red}{-50} \color{blue}{+50} \color{blue}{+75x} & = & \color{red}{-75x} +270 \color{blue}{+75x} \color{blue}{+50} \\\Leftrightarrow & 93x& = & 320 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{320}{93} \\\Leftrightarrow & x = \frac{320}{93} & & \\ & V = \left\{ \frac{320}{93} \right\} & \\\end{align}\)