Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 28x-252 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-28x} & = & \color{red}{28x} -252 \color{blue}{-28x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -228 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-228}{-7} \\\Leftrightarrow & x = \frac{228}{7} & & \\ & V = \left\{ \frac{228}{7} \right\} & \\\end{align}\)
2. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{364}{ \color{blue}{260} }x+\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-60& = & 364x+1560 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-364x} & = & \color{red}{364x} +1560 \color{blue}{-364x} \color{blue}{+60} \\\Leftrightarrow & -299x& = & 1620 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{1620}{-299} \\\Leftrightarrow & x = \frac{-1620}{299} & & \\ & V = \left\{ \frac{-1620}{299} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{7}{6}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & 245x-1260 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{-245x} & = & \color{red}{245x} -1260 \color{blue}{-245x} \color{blue}{+60} \\\Leftrightarrow & -203x& = & -1200 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{-1200}{-203} \\\Leftrightarrow & x = \frac{1200}{203} & & \\ & V = \left\{ \frac{1200}{203} \right\} & \\\end{align}\)
4. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }- \frac{ 28 }{ \color{blue}{63} })& = & (\frac{84}{ \color{blue}{63} }x+\frac{63}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x-28& = & 84x+63 \\\Leftrightarrow & 9x \color{red}{-28} \color{blue}{+28} \color{blue}{-84x} & = & \color{red}{84x} +63 \color{blue}{-84x} \color{blue}{+28} \\\Leftrightarrow & -75x& = & 91 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{91}{-75} \\\Leftrightarrow & x = \frac{-91}{75} & & \\ & V = \left\{ \frac{-91}{75} \right\} & \\\end{align}\)
5. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{5}{2}x-5 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{325}{ \color{blue}{130} }x-\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 325x-650 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-325x} & = & \color{red}{325x} -650 \color{blue}{-325x} \color{blue}{-40} \\\Leftrightarrow & -299x& = & -690 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{-690}{-299} \\\Leftrightarrow & x = \frac{30}{13} & & \\ & V = \left\{ \frac{30}{13} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & 140x-420 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{-140x} & = & \color{red}{140x} -420 \color{blue}{-140x} \color{blue}{-16} \\\Leftrightarrow & -125x& = & -436 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{-436}{-125} \\\Leftrightarrow & x = \frac{436}{125} & & \\ & V = \left\{ \frac{436}{125} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{12}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 35 }{ \color{blue}{84} })& = & (\frac{42}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+35& = & 42x+336 \\\Leftrightarrow & 12x \color{red}{+35} \color{blue}{-35} \color{blue}{-42x} & = & \color{red}{42x} +336 \color{blue}{-42x} \color{blue}{-35} \\\Leftrightarrow & -30x& = & 301 \\\Leftrightarrow & \frac{-30x}{ \color{red}{-30} }& = & \frac{301}{-30} \\\Leftrightarrow & x = \frac{-301}{30} & & \\ & V = \left\{ \frac{-301}{30} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -42x-180 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+42x} & = & \color{red}{-42x} -180 \color{blue}{+42x} \color{blue}{+9} \\\Leftrightarrow & 47x& = & -171 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-171}{47} \\\Leftrightarrow & x = \frac{-171}{47} & & \\ & V = \left\{ \frac{-171}{47} \right\} & \\\end{align}\)
9. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }- \frac{ 14 }{ \color{blue}{63} })& = & (\frac{-105}{ \color{blue}{63} }x-\frac{126}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x-14& = & -105x-126 \\\Leftrightarrow & 9x \color{red}{-14} \color{blue}{+14} \color{blue}{+105x} & = & \color{red}{-105x} -126 \color{blue}{+105x} \color{blue}{+14} \\\Leftrightarrow & 114x& = & -112 \\\Leftrightarrow & \frac{114x}{ \color{red}{114} }& = & \frac{-112}{114} \\\Leftrightarrow & x = \frac{-56}{57} & & \\ & V = \left\{ \frac{-56}{57} \right\} & \\\end{align}\)
10. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 10 }{ \color{blue}{45} })& = & (\frac{60}{ \color{blue}{45} }x-\frac{90}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-10& = & 60x-90 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{-60x} & = & \color{red}{60x} -90 \color{blue}{-60x} \color{blue}{+10} \\\Leftrightarrow & -51x& = & -80 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{-80}{-51} \\\Leftrightarrow & x = \frac{80}{51} & & \\ & V = \left\{ \frac{80}{51} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x-\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -96x-240 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+96x} & = & \color{red}{-96x} -240 \color{blue}{+96x} \color{blue}{-75} \\\Leftrightarrow & 136x& = & -315 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{-315}{136} \\\Leftrightarrow & x = \frac{-315}{136} & & \\ & V = \left\{ \frac{-315}{136} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+24& = & 21x+252 \\\Leftrightarrow & 28x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} +252 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & 7x& = & 228 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{228}{7} \\\Leftrightarrow & x = \frac{228}{7} & & \\ & V = \left\{ \frac{228}{7} \right\} & \\\end{align}\)