Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 4 }{ \color{blue}{15} })& = & (\frac{3}{ \color{blue}{15} }x-\frac{90}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+4& = & 3x-90 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-3x} & = & \color{red}{3x} -90 \color{blue}{-3x} \color{blue}{-4} \\\Leftrightarrow & 2x& = & -94 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{-94}{2} \\\Leftrightarrow & x = -47 & & \\ & V = \left\{ -47 \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{4}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 112x-672 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-112x} & = & \color{red}{112x} -672 \color{blue}{-112x} \color{blue}{-24} \\\Leftrightarrow & -91x& = & -696 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-696}{-91} \\\Leftrightarrow & x = \frac{696}{91} & & \\ & V = \left\{ \frac{696}{91} \right\} & \\\end{align}\)
3. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 20 }{ \color{blue}{35} })& = & (\frac{7}{ \color{blue}{35} }x+\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-20& = & 7x+70 \\\Leftrightarrow & 5x \color{red}{-20} \color{blue}{+20} \color{blue}{-7x} & = & \color{red}{7x} +70 \color{blue}{-7x} \color{blue}{+20} \\\Leftrightarrow & -2x& = & 90 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{90}{-2} \\\Leftrightarrow & x = -45 & & \\ & V = \left\{ -45 \right\} & \\\end{align}\)
4. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 70 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{182}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-70& = & 91x-182 \\\Leftrightarrow & 26x \color{red}{-70} \color{blue}{+70} \color{blue}{-91x} & = & \color{red}{91x} -182 \color{blue}{-91x} \color{blue}{+70} \\\Leftrightarrow & -65x& = & -112 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-112}{-65} \\\Leftrightarrow & x = \frac{112}{65} & & \\ & V = \left\{ \frac{112}{65} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{5}{3}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{50}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+9& = & 50x+240 \\\Leftrightarrow & 15x \color{red}{+9} \color{blue}{-9} \color{blue}{-50x} & = & \color{red}{50x} +240 \color{blue}{-50x} \color{blue}{-9} \\\Leftrightarrow & -35x& = & 231 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{231}{-35} \\\Leftrightarrow & x = \frac{-33}{5} & & \\ & V = \left\{ \frac{-33}{5} \right\} & \\\end{align}\)
6. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 7x+98 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-7x} & = & \color{red}{7x} +98 \color{blue}{-7x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 102 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{102}{-5} \\\Leftrightarrow & x = \frac{-102}{5} & & \\ & V = \left\{ \frac{-102}{5} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 50 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+50& = & 72x+360 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{-72x} & = & \color{red}{72x} +360 \color{blue}{-72x} \color{blue}{-50} \\\Leftrightarrow & -57x& = & 310 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{310}{-57} \\\Leftrightarrow & x = \frac{-310}{57} & & \\ & V = \left\{ \frac{-310}{57} \right\} & \\\end{align}\)
8. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{5}{3}x+2 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{75}{ \color{blue}{45} }x+\frac{90}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-20& = & 75x+90 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{-75x} & = & \color{red}{75x} +90 \color{blue}{-75x} \color{blue}{+20} \\\Leftrightarrow & -66x& = & 110 \\\Leftrightarrow & \frac{-66x}{ \color{red}{-66} }& = & \frac{110}{-66} \\\Leftrightarrow & x = \frac{-5}{3} & & \\ & V = \left\{ \frac{-5}{3} \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{4}{5}x-4 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{84}{ \color{blue}{105} }x-\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+30& = & 84x-420 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-84x} & = & \color{red}{84x} -420 \color{blue}{-84x} \color{blue}{-30} \\\Leftrightarrow & -49x& = & -450 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-450}{-49} \\\Leftrightarrow & x = \frac{450}{49} & & \\ & V = \left\{ \frac{450}{49} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -42x+240 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} +240 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 47x& = & 248 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{248}{47} \\\Leftrightarrow & x = \frac{248}{47} & & \\ & V = \left\{ \frac{248}{47} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{2}{5}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{24}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & 24x-420 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-24x} & = & \color{red}{24x} -420 \color{blue}{-24x} \color{blue}{-25} \\\Leftrightarrow & -14x& = & -445 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{-445}{-14} \\\Leftrightarrow & x = \frac{445}{14} & & \\ & V = \left\{ \frac{445}{14} \right\} & \\\end{align}\)
12. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 140 }{ \color{blue}{385} })& = & (\frac{-154}{ \color{blue}{385} }x+\frac{3080}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-140& = & -154x+3080 \\\Leftrightarrow & 55x \color{red}{-140} \color{blue}{+140} \color{blue}{+154x} & = & \color{red}{-154x} +3080 \color{blue}{+154x} \color{blue}{+140} \\\Leftrightarrow & 209x& = & 3220 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{3220}{209} \\\Leftrightarrow & x = \frac{3220}{209} & & \\ & V = \left\{ \frac{3220}{209} \right\} & \\\end{align}\)