Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}-\frac{2}{9}=\frac{3}{5}x+8\)
2. \(\frac{x}{3}-\frac{2}{7}=\frac{1}{2}x+3\)
3. \(\frac{x}{7}+\frac{5}{11}=\frac{-5}{6}x+3\)
4. \(\frac{x}{2}+\frac{5}{12}=\frac{-5}{3}x-8\)
5. \(\frac{x}{5}+\frac{2}{7}=\frac{1}{6}x-4\)
6. \(\frac{x}{4}-\frac{3}{14}=\frac{2}{3}x+2\)
7. \(\frac{x}{3}+\frac{2}{13}=\frac{1}{2}x-2\)
8. \(\frac{x}{6}-\frac{4}{15}=\frac{1}{5}x+8\)
9. \(\frac{x}{7}-\frac{3}{11}=\frac{-5}{6}x+7\)
10. \(\frac{x}{6}+\frac{4}{9}=\frac{-2}{5}x+3\)
11. \(\frac{x}{3}+\frac{3}{10}=\frac{7}{4}x-7\)
12. \(\frac{x}{5}+\frac{4}{15}=\frac{1}{6}x+1\)

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{3}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{54}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & 54x+720 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{-54x} & = & \color{red}{54x} +720 \color{blue}{-54x} \color{blue}{+20} \\\Leftrightarrow & -9x& = & 740 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{740}{-9} \\\Leftrightarrow & x = \frac{-740}{9} & & \\ & V = \left\{ \frac{-740}{9} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x+126 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} +126 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & 138 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{138}{-7} \\\Leftrightarrow & x = \frac{-138}{7} & & \\ & V = \left\{ \frac{-138}{7} \right\} & \\\end{align}\)
3. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 210 }{ \color{blue}{462} })& = & (\frac{-385}{ \color{blue}{462} }x+\frac{1386}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+210& = & -385x+1386 \\\Leftrightarrow & 66x \color{red}{+210} \color{blue}{-210} \color{blue}{+385x} & = & \color{red}{-385x} +1386 \color{blue}{+385x} \color{blue}{-210} \\\Leftrightarrow & 451x& = & 1176 \\\Leftrightarrow & \frac{451x}{ \color{red}{451} }& = & \frac{1176}{451} \\\Leftrightarrow & x = \frac{1176}{451} & & \\ & V = \left\{ \frac{1176}{451} \right\} & \\\end{align}\)
4. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{12}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-20}{ \color{blue}{12} }x-\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x+5& = & -20x-96 \\\Leftrightarrow & 6x \color{red}{+5} \color{blue}{-5} \color{blue}{+20x} & = & \color{red}{-20x} -96 \color{blue}{+20x} \color{blue}{-5} \\\Leftrightarrow & 26x& = & -101 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-101}{26} \\\Leftrightarrow & x = \frac{-101}{26} & & \\ & V = \left\{ \frac{-101}{26} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 35x-840 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-35x} & = & \color{red}{35x} -840 \color{blue}{-35x} \color{blue}{-60} \\\Leftrightarrow & 7x& = & -900 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-900}{7} \\\Leftrightarrow & x = \frac{-900}{7} & & \\ & V = \left\{ \frac{-900}{7} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & 56x+168 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-56x} & = & \color{red}{56x} +168 \color{blue}{-56x} \color{blue}{+18} \\\Leftrightarrow & -35x& = & 186 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{186}{-35} \\\Leftrightarrow & x = \frac{-186}{35} & & \\ & V = \left\{ \frac{-186}{35} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 39x-156 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-39x} & = & \color{red}{39x} -156 \color{blue}{-39x} \color{blue}{-12} \\\Leftrightarrow & -13x& = & -168 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-168}{-13} \\\Leftrightarrow & x = \frac{168}{13} & & \\ & V = \left\{ \frac{168}{13} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 6x+240 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-6x} & = & \color{red}{6x} +240 \color{blue}{-6x} \color{blue}{+8} \\\Leftrightarrow & -x& = & 248 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{248}{-1} \\\Leftrightarrow & x = -248 & & \\ & V = \left\{ -248 \right\} & \\\end{align}\)
9. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 126 }{ \color{blue}{462} })& = & (\frac{-385}{ \color{blue}{462} }x+\frac{3234}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-126& = & -385x+3234 \\\Leftrightarrow & 66x \color{red}{-126} \color{blue}{+126} \color{blue}{+385x} & = & \color{red}{-385x} +3234 \color{blue}{+385x} \color{blue}{+126} \\\Leftrightarrow & 451x& = & 3360 \\\Leftrightarrow & \frac{451x}{ \color{red}{451} }& = & \frac{3360}{451} \\\Leftrightarrow & x = \frac{3360}{451} & & \\ & V = \left\{ \frac{3360}{451} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -36x+270 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+36x} & = & \color{red}{-36x} +270 \color{blue}{+36x} \color{blue}{-40} \\\Leftrightarrow & 51x& = & 230 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{230}{51} \\\Leftrightarrow & x = \frac{230}{51} & & \\ & V = \left\{ \frac{230}{51} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{7}{4}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{105}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+18& = & 105x-420 \\\Leftrightarrow & 20x \color{red}{+18} \color{blue}{-18} \color{blue}{-105x} & = & \color{red}{105x} -420 \color{blue}{-105x} \color{blue}{-18} \\\Leftrightarrow & -85x& = & -438 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{-438}{-85} \\\Leftrightarrow & x = \frac{438}{85} & & \\ & V = \left\{ \frac{438}{85} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 5x+30 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-5x} & = & \color{red}{5x} +30 \color{blue}{-5x} \color{blue}{-8} \\\Leftrightarrow & x& = & 22 \\ & V = \left\{ 22 \right\} & \\\end{align}\)