Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-18& = & 15x+120 \\\Leftrightarrow & 20x \color{red}{-18} \color{blue}{+18} \color{blue}{-15x} & = & \color{red}{15x} +120 \color{blue}{-15x} \color{blue}{+18} \\\Leftrightarrow & 5x& = & 138 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{138}{5} \\\Leftrightarrow & x = \frac{138}{5} & & \\ & V = \left\{ \frac{138}{5} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 105x+126 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-105x} & = & \color{red}{105x} +126 \color{blue}{-105x} \color{blue}{-12} \\\Leftrightarrow & -91x& = & 114 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{114}{-91} \\\Leftrightarrow & x = \frac{-114}{91} & & \\ & V = \left\{ \frac{-114}{91} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-24& = & 33x+330 \\\Leftrightarrow & 22x \color{red}{-24} \color{blue}{+24} \color{blue}{-33x} & = & \color{red}{33x} +330 \color{blue}{-33x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & 354 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{354}{-11} \\\Leftrightarrow & x = \frac{-354}{11} & & \\ & V = \left\{ \frac{-354}{11} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{-88}{ \color{blue}{220} }x+\frac{1760}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-40& = & -88x+1760 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{+88x} & = & \color{red}{-88x} +1760 \color{blue}{+88x} \color{blue}{+40} \\\Leftrightarrow & 143x& = & 1800 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{1800}{143} \\\Leftrightarrow & x = \frac{1800}{143} & & \\ & V = \left\{ \frac{1800}{143} \right\} & \\\end{align}\)
5. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }+ \frac{ 10 }{ \color{blue}{45} })& = & (\frac{-120}{ \color{blue}{45} }x+\frac{225}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x+10& = & -120x+225 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{+120x} & = & \color{red}{-120x} +225 \color{blue}{+120x} \color{blue}{-10} \\\Leftrightarrow & 129x& = & 215 \\\Leftrightarrow & \frac{129x}{ \color{red}{129} }& = & \frac{215}{129} \\\Leftrightarrow & x = \frac{5}{3} & & \\ & V = \left\{ \frac{5}{3} \right\} & \\\end{align}\)
6. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{7}{5}x-1 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 28 }{ \color{blue}{105} })& = & (\frac{147}{ \color{blue}{105} }x-\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-28& = & 147x-105 \\\Leftrightarrow & 15x \color{red}{-28} \color{blue}{+28} \color{blue}{-147x} & = & \color{red}{147x} -105 \color{blue}{-147x} \color{blue}{+28} \\\Leftrightarrow & -132x& = & -77 \\\Leftrightarrow & \frac{-132x}{ \color{red}{-132} }& = & \frac{-77}{-132} \\\Leftrightarrow & x = \frac{7}{12} & & \\ & V = \left\{ \frac{7}{12} \right\} & \\\end{align}\)
7. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 21 }{ \color{blue}{56} })& = & (\frac{28}{ \color{blue}{56} }x+\frac{56}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+21& = & 28x+56 \\\Leftrightarrow & 8x \color{red}{+21} \color{blue}{-21} \color{blue}{-28x} & = & \color{red}{28x} +56 \color{blue}{-28x} \color{blue}{-21} \\\Leftrightarrow & -20x& = & 35 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{35}{-20} \\\Leftrightarrow & x = \frac{-7}{4} & & \\ & V = \left\{ \frac{-7}{4} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & 35x-840 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{-35x} & = & \color{red}{35x} -840 \color{blue}{-35x} \color{blue}{+90} \\\Leftrightarrow & 7x& = & -750 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-750}{7} \\\Leftrightarrow & x = \frac{-750}{7} & & \\ & V = \left\{ \frac{-750}{7} \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x-\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & -24x-288 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+24x} & = & \color{red}{-24x} -288 \color{blue}{+24x} \color{blue}{-8} \\\Leftrightarrow & 33x& = & -296 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-296}{33} \\\Leftrightarrow & x = \frac{-296}{33} & & \\ & V = \left\{ \frac{-296}{33} \right\} & \\\end{align}\)
10. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{7}{2}x+4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x+\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 49x+56 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-49x} & = & \color{red}{49x} +56 \color{blue}{-49x} \color{blue}{+4} \\\Leftrightarrow & -47x& = & 60 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{60}{-47} \\\Leftrightarrow & x = \frac{-60}{47} & & \\ & V = \left\{ \frac{-60}{47} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & -42x+90 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+42x} & = & \color{red}{-42x} +90 \color{blue}{+42x} \color{blue}{-4} \\\Leftrightarrow & 47x& = & 86 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{86}{47} \\\Leftrightarrow & x = \frac{86}{47} & & \\ & V = \left\{ \frac{86}{47} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{2}{3}x-7 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{70}{ \color{blue}{105} }x-\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+60& = & 70x-735 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{-70x} & = & \color{red}{70x} -735 \color{blue}{-70x} \color{blue}{-60} \\\Leftrightarrow & -49x& = & -795 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-795}{-49} \\\Leftrightarrow & x = \frac{795}{49} & & \\ & V = \left\{ \frac{795}{49} \right\} & \\\end{align}\)