Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-8}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -112x+336 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+112x} & = & \color{red}{-112x} +336 \color{blue}{+112x} \color{blue}{+24} \\\Leftrightarrow & 133x& = & 360 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{360}{133} \\\Leftrightarrow & x = \frac{360}{133} & & \\ & V = \left\{ \frac{360}{133} \right\} & \\\end{align}\)
2. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{-154}{ \color{blue}{110} }x-\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-50& = & -154x-660 \\\Leftrightarrow & 55x \color{red}{-50} \color{blue}{+50} \color{blue}{+154x} & = & \color{red}{-154x} -660 \color{blue}{+154x} \color{blue}{+50} \\\Leftrightarrow & 209x& = & -610 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-610}{209} \\\Leftrightarrow & x = \frac{-610}{209} & & \\ & V = \left\{ \frac{-610}{209} \right\} & \\\end{align}\)
3. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-24& = & -220x+264 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+220x} & = & \color{red}{-220x} +264 \color{blue}{+220x} \color{blue}{+24} \\\Leftrightarrow & 253x& = & 288 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{288}{253} \\\Leftrightarrow & x = \frac{288}{253} & & \\ & V = \left\{ \frac{288}{253} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{5}{3}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{30}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 30x-36 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-30x} & = & \color{red}{30x} -36 \color{blue}{-30x} \color{blue}{-4} \\\Leftrightarrow & -21x& = & -40 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-40}{-21} \\\Leftrightarrow & x = \frac{40}{21} & & \\ & V = \left\{ \frac{40}{21} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 468x-1950 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-468x} & = & \color{red}{468x} -1950 \color{blue}{-468x} \color{blue}{+120} \\\Leftrightarrow & -403x& = & -1830 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-1830}{-403} \\\Leftrightarrow & x = \frac{1830}{403} & & \\ & V = \left\{ \frac{1830}{403} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{234}{ \color{blue}{390} }x+\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & 234x+2730 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{-234x} & = & \color{red}{234x} +2730 \color{blue}{-234x} \color{blue}{-150} \\\Leftrightarrow & -169x& = & 2580 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{2580}{-169} \\\Leftrightarrow & x = \frac{-2580}{169} & & \\ & V = \left\{ \frac{-2580}{169} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+8& = & 75x-240 \\\Leftrightarrow & 10x \color{red}{+8} \color{blue}{-8} \color{blue}{-75x} & = & \color{red}{75x} -240 \color{blue}{-75x} \color{blue}{-8} \\\Leftrightarrow & -65x& = & -248 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-248}{-65} \\\Leftrightarrow & x = \frac{248}{65} & & \\ & V = \left\{ \frac{248}{65} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{9}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-20& = & -36x+180 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{+36x} & = & \color{red}{-36x} +180 \color{blue}{+36x} \color{blue}{+20} \\\Leftrightarrow & 51x& = & 200 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{200}{51} \\\Leftrightarrow & x = \frac{200}{51} & & \\ & V = \left\{ \frac{200}{51} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{84}{ \color{blue}{140} }x-\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 84x-840 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-84x} & = & \color{red}{84x} -840 \color{blue}{-84x} \color{blue}{-40} \\\Leftrightarrow & -49x& = & -880 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-880}{-49} \\\Leftrightarrow & x = \frac{880}{49} & & \\ & V = \left\{ \frac{880}{49} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-30& = & -28x+350 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{+28x} & = & \color{red}{-28x} +350 \color{blue}{+28x} \color{blue}{+30} \\\Leftrightarrow & 63x& = & 380 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{380}{63} \\\Leftrightarrow & x = \frac{380}{63} & & \\ & V = \left\{ \frac{380}{63} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{9}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+50& = & 45x+180 \\\Leftrightarrow & 18x \color{red}{+50} \color{blue}{-50} \color{blue}{-45x} & = & \color{red}{45x} +180 \color{blue}{-45x} \color{blue}{-50} \\\Leftrightarrow & -27x& = & 130 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{130}{-27} \\\Leftrightarrow & x = \frac{-130}{27} & & \\ & V = \left\{ \frac{-130}{27} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{26}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+30& = & 26x+650 \\\Leftrightarrow & 65x \color{red}{+30} \color{blue}{-30} \color{blue}{-26x} & = & \color{red}{26x} +650 \color{blue}{-26x} \color{blue}{-30} \\\Leftrightarrow & 39x& = & 620 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{620}{39} \\\Leftrightarrow & x = \frac{620}{39} & & \\ & V = \left\{ \frac{620}{39} \right\} & \\\end{align}\)