Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{7}{2}x+6 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{273}{ \color{blue}{78} }x+\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+24& = & 273x+468 \\\Leftrightarrow & 26x \color{red}{+24} \color{blue}{-24} \color{blue}{-273x} & = & \color{red}{273x} +468 \color{blue}{-273x} \color{blue}{-24} \\\Leftrightarrow & -247x& = & 444 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{444}{-247} \\\Leftrightarrow & x = \frac{-444}{247} & & \\ & V = \left\{ \frac{-444}{247} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{320}{ \color{blue}{240} }x-\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+45& = & 320x-1200 \\\Leftrightarrow & 48x \color{red}{+45} \color{blue}{-45} \color{blue}{-320x} & = & \color{red}{320x} -1200 \color{blue}{-320x} \color{blue}{-45} \\\Leftrightarrow & -272x& = & -1245 \\\Leftrightarrow & \frac{-272x}{ \color{red}{-272} }& = & \frac{-1245}{-272} \\\Leftrightarrow & x = \frac{1245}{272} & & \\ & V = \left\{ \frac{1245}{272} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & -312x+1560 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{+312x} & = & \color{red}{-312x} +1560 \color{blue}{+312x} \color{blue}{+90} \\\Leftrightarrow & 377x& = & 1650 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{1650}{377} \\\Leftrightarrow & x = \frac{1650}{377} & & \\ & V = \left\{ \frac{1650}{377} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{9}& = & \frac{7}{2}x-4 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{315}{ \color{blue}{90} }x-\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-50& = & 315x-360 \\\Leftrightarrow & 18x \color{red}{-50} \color{blue}{+50} \color{blue}{-315x} & = & \color{red}{315x} -360 \color{blue}{-315x} \color{blue}{+50} \\\Leftrightarrow & -297x& = & -310 \\\Leftrightarrow & \frac{-297x}{ \color{red}{-297} }& = & \frac{-310}{-297} \\\Leftrightarrow & x = \frac{310}{297} & & \\ & V = \left\{ \frac{310}{297} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x+210 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} +210 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & 190 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{190}{-21} \\\Leftrightarrow & x = \frac{-190}{21} & & \\ & V = \left\{ \frac{-190}{21} \right\} & \\\end{align}\)
6. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x+\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-60& = & -105x+1120 \\\Leftrightarrow & 28x \color{red}{-60} \color{blue}{+60} \color{blue}{+105x} & = & \color{red}{-105x} +1120 \color{blue}{+105x} \color{blue}{+60} \\\Leftrightarrow & 133x& = & 1180 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{1180}{133} \\\Leftrightarrow & x = \frac{1180}{133} & & \\ & V = \left\{ \frac{1180}{133} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x-\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & -12x-108 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+12x} & = & \color{red}{-12x} -108 \color{blue}{+12x} \color{blue}{-8} \\\Leftrightarrow & 21x& = & -116 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-116}{21} \\\Leftrightarrow & x = \frac{-116}{21} & & \\ & V = \left\{ \frac{-116}{21} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+9& = & 5x-210 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{-5x} & = & \color{red}{5x} -210 \color{blue}{-5x} \color{blue}{-9} \\\Leftrightarrow & x& = & -219 \\ & V = \left\{ -219 \right\} & \\\end{align}\)
9. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{4}{5}x+3 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{144}{ \color{blue}{180} }x+\frac{540}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+80& = & 144x+540 \\\Leftrightarrow & 45x \color{red}{+80} \color{blue}{-80} \color{blue}{-144x} & = & \color{red}{144x} +540 \color{blue}{-144x} \color{blue}{-80} \\\Leftrightarrow & -99x& = & 460 \\\Leftrightarrow & \frac{-99x}{ \color{red}{-99} }& = & \frac{460}{-99} \\\Leftrightarrow & x = \frac{-460}{99} & & \\ & V = \left\{ \frac{-460}{99} \right\} & \\\end{align}\)
10. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{168.} (\frac{24x}{ \color{blue}{168} }+ \frac{ 63 }{ \color{blue}{168} })& = & (\frac{392}{ \color{blue}{168} }x+\frac{1176}{ \color{blue}{168} }) \color{blue}{.168} \\\Leftrightarrow & 24x+63& = & 392x+1176 \\\Leftrightarrow & 24x \color{red}{+63} \color{blue}{-63} \color{blue}{-392x} & = & \color{red}{392x} +1176 \color{blue}{-392x} \color{blue}{-63} \\\Leftrightarrow & -368x& = & 1113 \\\Leftrightarrow & \frac{-368x}{ \color{red}{-368} }& = & \frac{1113}{-368} \\\Leftrightarrow & x = \frac{-1113}{368} & & \\ & V = \left\{ \frac{-1113}{368} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & -130x-234 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+130x} & = & \color{red}{-130x} -234 \color{blue}{+130x} \color{blue}{+24} \\\Leftrightarrow & 169x& = & -210 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-210}{169} \\\Leftrightarrow & x = \frac{-210}{169} & & \\ & V = \left\{ \frac{-210}{169} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 66x-1980 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-66x} & = & \color{red}{66x} -1980 \color{blue}{-66x} \color{blue}{+150} \\\Leftrightarrow & -11x& = & -1830 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-1830}{-11} \\\Leftrightarrow & x = \frac{1830}{11} & & \\ & V = \left\{ \frac{1830}{11} \right\} & \\\end{align}\)