Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}+\frac{4}{9}=\frac{1}{2}x+6\)
2. \(\frac{x}{2}+\frac{2}{13}=\frac{4}{3}x+1\)
3. \(\frac{x}{6}-\frac{3}{16}=\frac{6}{5}x+7\)
4. \(\frac{x}{2}-\frac{5}{7}=\frac{-2}{5}x-1\)
5. \(\frac{x}{3}-\frac{4}{15}=\frac{1}{2}x-5\)
6. \(\frac{x}{3}-\frac{2}{15}=\frac{-7}{5}x-7\)
7. \(\frac{x}{3}-\frac{5}{11}=\frac{1}{2}x+8\)
8. \(\frac{x}{4}-\frac{5}{11}=\frac{1}{3}x-5\)
9. \(\frac{x}{5}+\frac{2}{7}=\frac{7}{2}x-1\)
10. \(\frac{x}{2}-\frac{5}{8}=\frac{-5}{3}x+1\)
11. \(\frac{x}{2}+\frac{4}{9}=\frac{2}{3}x-1\)
12. \(\frac{x}{2}-\frac{5}{16}=\frac{-5}{3}x-7\)

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 45x+540 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-45x} & = & \color{red}{45x} +540 \color{blue}{-45x} \color{blue}{-40} \\\Leftrightarrow & -27x& = & 500 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{500}{-27} \\\Leftrightarrow & x = \frac{-500}{27} & & \\ & V = \left\{ \frac{-500}{27} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{104}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & 104x+78 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{-104x} & = & \color{red}{104x} +78 \color{blue}{-104x} \color{blue}{-12} \\\Leftrightarrow & -65x& = & 66 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{66}{-65} \\\Leftrightarrow & x = \frac{-66}{65} & & \\ & V = \left\{ \frac{-66}{65} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 288x+1680 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-288x} & = & \color{red}{288x} +1680 \color{blue}{-288x} \color{blue}{+45} \\\Leftrightarrow & -248x& = & 1725 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{1725}{-248} \\\Leftrightarrow & x = \frac{-1725}{248} & & \\ & V = \left\{ \frac{-1725}{248} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 50 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x-\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-50& = & -28x-70 \\\Leftrightarrow & 35x \color{red}{-50} \color{blue}{+50} \color{blue}{+28x} & = & \color{red}{-28x} -70 \color{blue}{+28x} \color{blue}{+50} \\\Leftrightarrow & 63x& = & -20 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{-20}{63} \\\Leftrightarrow & x = \frac{-20}{63} & & \\ & V = \left\{ \frac{-20}{63} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-8& = & 15x-150 \\\Leftrightarrow & 10x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} -150 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -142 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-142}{-5} \\\Leftrightarrow & x = \frac{142}{5} & & \\ & V = \left\{ \frac{142}{5} \right\} & \\\end{align}\)
6. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }- \frac{ 2 }{ \color{blue}{15} })& = & (\frac{-21}{ \color{blue}{15} }x-\frac{105}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x-2& = & -21x-105 \\\Leftrightarrow & 5x \color{red}{-2} \color{blue}{+2} \color{blue}{+21x} & = & \color{red}{-21x} -105 \color{blue}{+21x} \color{blue}{+2} \\\Leftrightarrow & 26x& = & -103 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-103}{26} \\\Leftrightarrow & x = \frac{-103}{26} & & \\ & V = \left\{ \frac{-103}{26} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-30& = & 33x+528 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-33x} & = & \color{red}{33x} +528 \color{blue}{-33x} \color{blue}{+30} \\\Leftrightarrow & -11x& = & 558 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{558}{-11} \\\Leftrightarrow & x = \frac{-558}{11} & & \\ & V = \left\{ \frac{-558}{11} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-60& = & 44x-660 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{-44x} & = & \color{red}{44x} -660 \color{blue}{-44x} \color{blue}{+60} \\\Leftrightarrow & -11x& = & -600 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-600}{-11} \\\Leftrightarrow & x = \frac{600}{11} & & \\ & V = \left\{ \frac{600}{11} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{7}{2}x-1 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x-\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 245x-70 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-245x} & = & \color{red}{245x} -70 \color{blue}{-245x} \color{blue}{-20} \\\Leftrightarrow & -231x& = & -90 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{-90}{-231} \\\Leftrightarrow & x = \frac{30}{77} & & \\ & V = \left\{ \frac{30}{77} \right\} & \\\end{align}\)
10. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-40}{ \color{blue}{24} }x+\frac{24}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & -40x+24 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+40x} & = & \color{red}{-40x} +24 \color{blue}{+40x} \color{blue}{+15} \\\Leftrightarrow & 52x& = & 39 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{39}{52} \\\Leftrightarrow & x = \frac{3}{4} & & \\ & V = \left\{ \frac{3}{4} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{2}{3}x-1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{12}{ \color{blue}{18} }x-\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & 12x-18 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-12x} & = & \color{red}{12x} -18 \color{blue}{-12x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & -26 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-26}{-3} \\\Leftrightarrow & x = \frac{26}{3} & & \\ & V = \left\{ \frac{26}{3} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x-\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & -80x-336 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{+80x} & = & \color{red}{-80x} -336 \color{blue}{+80x} \color{blue}{+15} \\\Leftrightarrow & 104x& = & -321 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{-321}{104} \\\Leftrightarrow & x = \frac{-321}{104} & & \\ & V = \left\{ \frac{-321}{104} \right\} & \\\end{align}\)