Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{3}{8}=\frac{1}{2}x-4\)
2. \(\frac{x}{5}+\frac{5}{9}=\frac{-2}{3}x-5\)
3. \(\frac{x}{2}-\frac{3}{13}=\frac{-2}{5}x+3\)
4. \(\frac{x}{3}+\frac{5}{11}=\frac{-7}{4}x-3\)
5. \(\frac{x}{4}+\frac{4}{15}=\frac{1}{3}x-4\)
6. \(\frac{x}{5}-\frac{5}{16}=\frac{-5}{6}x+3\)
7. \(\frac{x}{5}+\frac{3}{13}=\frac{1}{2}x+8\)
8. \(\frac{x}{7}-\frac{4}{15}=\frac{3}{2}x-6\)
9. \(\frac{x}{2}+\frac{4}{15}=\frac{2}{5}x-4\)
10. \(\frac{x}{7}-\frac{3}{11}=\frac{-2}{3}x-1\)
11. \(\frac{x}{5}+\frac{4}{15}=\frac{1}{2}x+4\)
12. \(\frac{x}{7}-\frac{2}{15}=\frac{3}{2}x-3\)

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

Verbetersleutel

1. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 21 }{ \color{blue}{56} })& = & (\frac{28}{ \color{blue}{56} }x-\frac{224}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+21& = & 28x-224 \\\Leftrightarrow & 8x \color{red}{+21} \color{blue}{-21} \color{blue}{-28x} & = & \color{red}{28x} -224 \color{blue}{-28x} \color{blue}{-21} \\\Leftrightarrow & -20x& = & -245 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{-245}{-20} \\\Leftrightarrow & x = \frac{49}{4} & & \\ & V = \left\{ \frac{49}{4} \right\} & \\\end{align}\)
2. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{9}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }+ \frac{ 25 }{ \color{blue}{45} })& = & (\frac{-30}{ \color{blue}{45} }x-\frac{225}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x+25& = & -30x-225 \\\Leftrightarrow & 9x \color{red}{+25} \color{blue}{-25} \color{blue}{+30x} & = & \color{red}{-30x} -225 \color{blue}{+30x} \color{blue}{-25} \\\Leftrightarrow & 39x& = & -250 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-250}{39} \\\Leftrightarrow & x = \frac{-250}{39} & & \\ & V = \left\{ \frac{-250}{39} \right\} & \\\end{align}\)
3. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x+\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-30& = & -52x+390 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{+52x} & = & \color{red}{-52x} +390 \color{blue}{+52x} \color{blue}{+30} \\\Leftrightarrow & 117x& = & 420 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{420}{117} \\\Leftrightarrow & x = \frac{140}{39} & & \\ & V = \left\{ \frac{140}{39} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-231}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+60& = & -231x-396 \\\Leftrightarrow & 44x \color{red}{+60} \color{blue}{-60} \color{blue}{+231x} & = & \color{red}{-231x} -396 \color{blue}{+231x} \color{blue}{-60} \\\Leftrightarrow & 275x& = & -456 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{-456}{275} \\\Leftrightarrow & x = \frac{-456}{275} & & \\ & V = \left\{ \frac{-456}{275} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & 20x-240 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{-20x} & = & \color{red}{20x} -240 \color{blue}{-20x} \color{blue}{-16} \\\Leftrightarrow & -5x& = & -256 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-256}{-5} \\\Leftrightarrow & x = \frac{256}{5} & & \\ & V = \left\{ \frac{256}{5} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x+\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & -200x+720 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{+200x} & = & \color{red}{-200x} +720 \color{blue}{+200x} \color{blue}{+75} \\\Leftrightarrow & 248x& = & 795 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{795}{248} \\\Leftrightarrow & x = \frac{795}{248} & & \\ & V = \left\{ \frac{795}{248} \right\} & \\\end{align}\)
7. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x+\frac{1040}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x+1040 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} +1040 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & 1010 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{1010}{-39} \\\Leftrightarrow & x = \frac{-1010}{39} & & \\ & V = \left\{ \frac{-1010}{39} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 56 }{ \color{blue}{210} })& = & (\frac{315}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-56& = & 315x-1260 \\\Leftrightarrow & 30x \color{red}{-56} \color{blue}{+56} \color{blue}{-315x} & = & \color{red}{315x} -1260 \color{blue}{-315x} \color{blue}{+56} \\\Leftrightarrow & -285x& = & -1204 \\\Leftrightarrow & \frac{-285x}{ \color{red}{-285} }& = & \frac{-1204}{-285} \\\Leftrightarrow & x = \frac{1204}{285} & & \\ & V = \left\{ \frac{1204}{285} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{2}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 12x-120 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-12x} & = & \color{red}{12x} -120 \color{blue}{-12x} \color{blue}{-8} \\\Leftrightarrow & 3x& = & -128 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-128}{3} \\\Leftrightarrow & x = \frac{-128}{3} & & \\ & V = \left\{ \frac{-128}{3} \right\} & \\\end{align}\)
10. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 63 }{ \color{blue}{231} })& = & (\frac{-154}{ \color{blue}{231} }x-\frac{231}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-63& = & -154x-231 \\\Leftrightarrow & 33x \color{red}{-63} \color{blue}{+63} \color{blue}{+154x} & = & \color{red}{-154x} -231 \color{blue}{+154x} \color{blue}{+63} \\\Leftrightarrow & 187x& = & -168 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-168}{187} \\\Leftrightarrow & x = \frac{-168}{187} & & \\ & V = \left\{ \frac{-168}{187} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 15x+120 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-15x} & = & \color{red}{15x} +120 \color{blue}{-15x} \color{blue}{-8} \\\Leftrightarrow & -9x& = & 112 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{112}{-9} \\\Leftrightarrow & x = \frac{-112}{9} & & \\ & V = \left\{ \frac{-112}{9} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 28 }{ \color{blue}{210} })& = & (\frac{315}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-28& = & 315x-630 \\\Leftrightarrow & 30x \color{red}{-28} \color{blue}{+28} \color{blue}{-315x} & = & \color{red}{315x} -630 \color{blue}{-315x} \color{blue}{+28} \\\Leftrightarrow & -285x& = & -602 \\\Leftrightarrow & \frac{-285x}{ \color{red}{-285} }& = & \frac{-602}{-285} \\\Leftrightarrow & x = \frac{602}{285} & & \\ & V = \left\{ \frac{602}{285} \right\} & \\\end{align}\)