Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x-\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-15& = & 112x-288 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{-112x} & = & \color{red}{112x} -288 \color{blue}{-112x} \color{blue}{+15} \\\Leftrightarrow & -100x& = & -273 \\\Leftrightarrow & \frac{-100x}{ \color{red}{-100} }& = & \frac{-273}{-100} \\\Leftrightarrow & x = \frac{273}{100} & & \\ & V = \left\{ \frac{273}{100} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x-\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x-75& = & -192x-720 \\\Leftrightarrow & 80x \color{red}{-75} \color{blue}{+75} \color{blue}{+192x} & = & \color{red}{-192x} -720 \color{blue}{+192x} \color{blue}{+75} \\\Leftrightarrow & 272x& = & -645 \\\Leftrightarrow & \frac{272x}{ \color{red}{272} }& = & \frac{-645}{272} \\\Leftrightarrow & x = \frac{-645}{272} & & \\ & V = \left\{ \frac{-645}{272} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -42x-240 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} -240 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 47x& = & -232 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-232}{47} \\\Leftrightarrow & x = \frac{-232}{47} & & \\ & V = \left\{ \frac{-232}{47} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 3, 6 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-25& = & -24x+120 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{+24x} & = & \color{red}{-24x} +120 \color{blue}{+24x} \color{blue}{+25} \\\Leftrightarrow & 34x& = & 145 \\\Leftrightarrow & \frac{34x}{ \color{red}{34} }& = & \frac{145}{34} \\\Leftrightarrow & x = \frac{145}{34} & & \\ & V = \left\{ \frac{145}{34} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+24& = & 7x+210 \\\Leftrightarrow & 6x \color{red}{+24} \color{blue}{-24} \color{blue}{-7x} & = & \color{red}{7x} +210 \color{blue}{-7x} \color{blue}{-24} \\\Leftrightarrow & -x& = & 186 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{186}{-1} \\\Leftrightarrow & x = -186 & & \\ & V = \left\{ -186 \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & 176x+660 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{-176x} & = & \color{red}{176x} +660 \color{blue}{-176x} \color{blue}{-48} \\\Leftrightarrow & -143x& = & 612 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{612}{-143} \\\Leftrightarrow & x = \frac{-612}{143} & & \\ & V = \left\{ \frac{-612}{143} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & 20x-60 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{-20x} & = & \color{red}{20x} -60 \color{blue}{-20x} \color{blue}{-25} \\\Leftrightarrow & -8x& = & -85 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-85}{-8} \\\Leftrightarrow & x = \frac{85}{8} & & \\ & V = \left\{ \frac{85}{8} \right\} & \\\end{align}\)
8. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 50 }{ \color{blue}{130} })& = & (\frac{455}{ \color{blue}{130} }x+\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-50& = & 455x+910 \\\Leftrightarrow & 26x \color{red}{-50} \color{blue}{+50} \color{blue}{-455x} & = & \color{red}{455x} +910 \color{blue}{-455x} \color{blue}{+50} \\\Leftrightarrow & -429x& = & 960 \\\Leftrightarrow & \frac{-429x}{ \color{red}{-429} }& = & \frac{960}{-429} \\\Leftrightarrow & x = \frac{-320}{143} & & \\ & V = \left\{ \frac{-320}{143} \right\} & \\\end{align}\)
9. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{182}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 91x+182 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-91x} & = & \color{red}{91x} +182 \color{blue}{-91x} \color{blue}{+42} \\\Leftrightarrow & -65x& = & 224 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{224}{-65} \\\Leftrightarrow & x = \frac{-224}{65} & & \\ & V = \left\{ \frac{-224}{65} \right\} & \\\end{align}\)
10. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{5}{2}x+6 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{45}{ \color{blue}{18} }x+\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-4& = & 45x+108 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-45x} & = & \color{red}{45x} +108 \color{blue}{-45x} \color{blue}{+4} \\\Leftrightarrow & -39x& = & 112 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{112}{-39} \\\Leftrightarrow & x = \frac{-112}{39} & & \\ & V = \left\{ \frac{-112}{39} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{4}{5}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{112}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 112x+700 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-112x} & = & \color{red}{112x} +700 \color{blue}{-112x} \color{blue}{-80} \\\Leftrightarrow & -77x& = & 620 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{620}{-77} \\\Leftrightarrow & x = \frac{-620}{77} & & \\ & V = \left\{ \frac{-620}{77} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{12}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-48}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x-25& = & -48x-120 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{+48x} & = & \color{red}{-48x} -120 \color{blue}{+48x} \color{blue}{+25} \\\Leftrightarrow & 58x& = & -95 \\\Leftrightarrow & \frac{58x}{ \color{red}{58} }& = & \frac{-95}{58} \\\Leftrightarrow & x = \frac{-95}{58} & & \\ & V = \left\{ \frac{-95}{58} \right\} & \\\end{align}\)