Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{24}{ \color{blue}{120} }x-\frac{960}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 24x-960 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-24x} & = & \color{red}{24x} -960 \color{blue}{-24x} \color{blue}{+75} \\\Leftrightarrow & -4x& = & -885 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-885}{-4} \\\Leftrightarrow & x = \frac{885}{4} & & \\ & V = \left\{ \frac{885}{4} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 42x+630 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-42x} & = & \color{red}{42x} +630 \color{blue}{-42x} \color{blue}{+90} \\\Leftrightarrow & -7x& = & 720 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{720}{-7} \\\Leftrightarrow & x = \frac{-720}{7} & & \\ & V = \left\{ \frac{-720}{7} \right\} & \\\end{align}\)
3. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }- \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-273}{ \color{blue}{195} }x+\frac{1365}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x-45& = & -273x+1365 \\\Leftrightarrow & 65x \color{red}{-45} \color{blue}{+45} \color{blue}{+273x} & = & \color{red}{-273x} +1365 \color{blue}{+273x} \color{blue}{+45} \\\Leftrightarrow & 338x& = & 1410 \\\Leftrightarrow & \frac{338x}{ \color{red}{338} }& = & \frac{1410}{338} \\\Leftrightarrow & x = \frac{705}{169} & & \\ & V = \left\{ \frac{705}{169} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{7}{3}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & 140x-120 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{-140x} & = & \color{red}{140x} -120 \color{blue}{-140x} \color{blue}{-18} \\\Leftrightarrow & -125x& = & -138 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{-138}{-125} \\\Leftrightarrow & x = \frac{138}{125} & & \\ & V = \left\{ \frac{138}{125} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 15x-180 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-15x} & = & \color{red}{15x} -180 \color{blue}{-15x} \color{blue}{-25} \\\Leftrightarrow & -9x& = & -205 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-205}{-9} \\\Leftrightarrow & x = \frac{205}{9} & & \\ & V = \left\{ \frac{205}{9} \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & 176x+660 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-176x} & = & \color{red}{176x} +660 \color{blue}{-176x} \color{blue}{-24} \\\Leftrightarrow & -143x& = & 636 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{636}{-143} \\\Leftrightarrow & x = \frac{-636}{143} & & \\ & V = \left\{ \frac{-636}{143} \right\} & \\\end{align}\)
7. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{15}{ \color{blue}{10} }x-\frac{60}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x+3& = & 15x-60 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-15x} & = & \color{red}{15x} -60 \color{blue}{-15x} \color{blue}{-3} \\\Leftrightarrow & -13x& = & -63 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-63}{-13} \\\Leftrightarrow & x = \frac{63}{13} & & \\ & V = \left\{ \frac{63}{13} \right\} & \\\end{align}\)
8. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{13}& = & \frac{2}{5}x-5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 50 }{ \color{blue}{130} })& = & (\frac{52}{ \color{blue}{130} }x-\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+50& = & 52x-650 \\\Leftrightarrow & 65x \color{red}{+50} \color{blue}{-50} \color{blue}{-52x} & = & \color{red}{52x} -650 \color{blue}{-52x} \color{blue}{-50} \\\Leftrightarrow & 13x& = & -700 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-700}{13} \\\Leftrightarrow & x = \frac{-700}{13} & & \\ & V = \left\{ \frac{-700}{13} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -42x-240 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+42x} & = & \color{red}{-42x} -240 \color{blue}{+42x} \color{blue}{-9} \\\Leftrightarrow & 47x& = & -249 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-249}{47} \\\Leftrightarrow & x = \frac{-249}{47} & & \\ & V = \left\{ \frac{-249}{47} \right\} & \\\end{align}\)
10. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 25 }{ \color{blue}{35} })& = & (\frac{28}{ \color{blue}{35} }x+\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-25& = & 28x+70 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{-28x} & = & \color{red}{28x} +70 \color{blue}{-28x} \color{blue}{+25} \\\Leftrightarrow & -23x& = & 95 \\\Leftrightarrow & \frac{-23x}{ \color{red}{-23} }& = & \frac{95}{-23} \\\Leftrightarrow & x = \frac{-95}{23} & & \\ & V = \left\{ \frac{-95}{23} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x+\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & 396x+660 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{-396x} & = & \color{red}{396x} +660 \color{blue}{-396x} \color{blue}{-90} \\\Leftrightarrow & -341x& = & 570 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{570}{-341} \\\Leftrightarrow & x = \frac{-570}{341} & & \\ & V = \left\{ \frac{-570}{341} \right\} & \\\end{align}\)
12. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 42 }{ \color{blue}{231} })& = & (\frac{77}{ \color{blue}{231} }x-\frac{231}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+42& = & 77x-231 \\\Leftrightarrow & 33x \color{red}{+42} \color{blue}{-42} \color{blue}{-77x} & = & \color{red}{77x} -231 \color{blue}{-77x} \color{blue}{-42} \\\Leftrightarrow & -44x& = & -273 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-273}{-44} \\\Leftrightarrow & x = \frac{273}{44} & & \\ & V = \left\{ \frac{273}{44} \right\} & \\\end{align}\)