Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x-\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -192x-480 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+192x} & = & \color{red}{-192x} -480 \color{blue}{+192x} \color{blue}{-45} \\\Leftrightarrow & 232x& = & -525 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{-525}{232} \\\Leftrightarrow & x = \frac{-525}{232} & & \\ & V = \left\{ \frac{-525}{232} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x-\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-24& = & 176x-264 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{-176x} & = & \color{red}{176x} -264 \color{blue}{-176x} \color{blue}{+24} \\\Leftrightarrow & -143x& = & -240 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-240}{-143} \\\Leftrightarrow & x = \frac{240}{143} & & \\ & V = \left\{ \frac{240}{143} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 45 }{ \color{blue}{165} })& = & (\frac{198}{ \color{blue}{165} }x+\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-45& = & 198x+330 \\\Leftrightarrow & 55x \color{red}{-45} \color{blue}{+45} \color{blue}{-198x} & = & \color{red}{198x} +330 \color{blue}{-198x} \color{blue}{+45} \\\Leftrightarrow & -143x& = & 375 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{375}{-143} \\\Leftrightarrow & x = \frac{-375}{143} & & \\ & V = \left\{ \frac{-375}{143} \right\} & \\\end{align}\)
4. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{5}{2}x+1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x+\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 35x+14 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-35x} & = & \color{red}{35x} +14 \color{blue}{-35x} \color{blue}{-4} \\\Leftrightarrow & -33x& = & 10 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{10}{-33} \\\Leftrightarrow & x = \frac{-10}{33} & & \\ & V = \left\{ \frac{-10}{33} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 2, 12 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{12}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{30x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 30x-25& = & -84x-240 \\\Leftrightarrow & 30x \color{red}{-25} \color{blue}{+25} \color{blue}{+84x} & = & \color{red}{-84x} -240 \color{blue}{+84x} \color{blue}{+25} \\\Leftrightarrow & 114x& = & -215 \\\Leftrightarrow & \frac{114x}{ \color{red}{114} }& = & \frac{-215}{114} \\\Leftrightarrow & x = \frac{-215}{114} & & \\ & V = \left\{ \frac{-215}{114} \right\} & \\\end{align}\)
6. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{36}{ \color{blue}{180} }x+\frac{360}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+40& = & 36x+360 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{-36x} & = & \color{red}{36x} +360 \color{blue}{-36x} \color{blue}{-40} \\\Leftrightarrow & 9x& = & 320 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{320}{9} \\\Leftrightarrow & x = \frac{320}{9} & & \\ & V = \left\{ \frac{320}{9} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & -100x-420 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{+100x} & = & \color{red}{-100x} -420 \color{blue}{+100x} \color{blue}{+18} \\\Leftrightarrow & 115x& = & -402 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{-402}{115} \\\Leftrightarrow & x = \frac{-402}{115} & & \\ & V = \left\{ \frac{-402}{115} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+24& = & 39x-468 \\\Leftrightarrow & 26x \color{red}{+24} \color{blue}{-24} \color{blue}{-39x} & = & \color{red}{39x} -468 \color{blue}{-39x} \color{blue}{-24} \\\Leftrightarrow & -13x& = & -492 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-492}{-13} \\\Leftrightarrow & x = \frac{492}{13} & & \\ & V = \left\{ \frac{492}{13} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & -25x+90 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{+25x} & = & \color{red}{-25x} +90 \color{blue}{+25x} \color{blue}{+9} \\\Leftrightarrow & 31x& = & 99 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{99}{31} \\\Leftrightarrow & x = \frac{99}{31} & & \\ & V = \left\{ \frac{99}{31} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -84x-630 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} -630 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & -690 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-690}{119} \\\Leftrightarrow & x = \frac{-690}{119} & & \\ & V = \left\{ \frac{-690}{119} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{1}{4}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-18& = & 15x+420 \\\Leftrightarrow & 20x \color{red}{-18} \color{blue}{+18} \color{blue}{-15x} & = & \color{red}{15x} +420 \color{blue}{-15x} \color{blue}{+18} \\\Leftrightarrow & 5x& = & 438 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{438}{5} \\\Leftrightarrow & x = \frac{438}{5} & & \\ & V = \left\{ \frac{438}{5} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -84x+1050 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+84x} & = & \color{red}{-84x} +1050 \color{blue}{+84x} \color{blue}{-120} \\\Leftrightarrow & 119x& = & 930 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{930}{119} \\\Leftrightarrow & x = \frac{930}{119} & & \\ & V = \left\{ \frac{930}{119} \right\} & \\\end{align}\)