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. \(\frac{x}{3}+\frac{3}{13}=\frac{5}{4}x-1\)
2. \(\frac{x}{3}+\frac{3}{14}=\frac{1}{4}x-1\)
3. \(\frac{x}{5}+\frac{2}{13}=\frac{7}{2}x+3\)
4. \(\frac{x}{7}-\frac{2}{15}=\frac{1}{6}x+3\)
5. \(\frac{x}{5}-\frac{4}{15}=\frac{1}{2}x-7\)
6. \(\frac{x}{4}+\frac{2}{7}=\frac{4}{3}x+2\)
7. \(\frac{x}{7}+\frac{3}{7}=\frac{-5}{3}x-6\)
8. \(\frac{x}{6}+\frac{5}{13}=\frac{-2}{5}x+4\)
9. \(\frac{x}{3}-\frac{4}{11}=\frac{-3}{4}x+3\)
10. \(\frac{x}{4}+\frac{5}{16}=\frac{3}{5}x+5\)
11. \(\frac{x}{6}-\frac{5}{8}=\frac{6}{5}x+1\)
12. \(\frac{x}{4}-\frac{4}{11}=\frac{-5}{3}x+6\)

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{5}{4}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{195}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+36& = & 195x-156 \\\Leftrightarrow & 52x \color{red}{+36} \color{blue}{-36} \color{blue}{-195x} & = & \color{red}{195x} -156 \color{blue}{-195x} \color{blue}{-36} \\\Leftrightarrow & -143x& = & -192 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-192}{-143} \\\Leftrightarrow & x = \frac{192}{143} & & \\ & V = \left\{ \frac{192}{143} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{14}& = & \frac{1}{4}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 18 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+18& = & 21x-84 \\\Leftrightarrow & 28x \color{red}{+18} \color{blue}{-18} \color{blue}{-21x} & = & \color{red}{21x} -84 \color{blue}{-21x} \color{blue}{-18} \\\Leftrightarrow & 7x& = & -102 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-102}{7} \\\Leftrightarrow & x = \frac{-102}{7} & & \\ & V = \left\{ \frac{-102}{7} \right\} & \\\end{align}\)
3. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{7}{2}x+3 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{455}{ \color{blue}{130} }x+\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+20& = & 455x+390 \\\Leftrightarrow & 26x \color{red}{+20} \color{blue}{-20} \color{blue}{-455x} & = & \color{red}{455x} +390 \color{blue}{-455x} \color{blue}{-20} \\\Leftrightarrow & -429x& = & 370 \\\Leftrightarrow & \frac{-429x}{ \color{red}{-429} }& = & \frac{370}{-429} \\\Leftrightarrow & x = \frac{-370}{429} & & \\ & V = \left\{ \frac{-370}{429} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 28 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-28& = & 35x+630 \\\Leftrightarrow & 30x \color{red}{-28} \color{blue}{+28} \color{blue}{-35x} & = & \color{red}{35x} +630 \color{blue}{-35x} \color{blue}{+28} \\\Leftrightarrow & -5x& = & 658 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{658}{-5} \\\Leftrightarrow & x = \frac{-658}{5} & & \\ & V = \left\{ \frac{-658}{5} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 15x-210 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} -210 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -9x& = & -202 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-202}{-9} \\\Leftrightarrow & x = \frac{202}{9} & & \\ & V = \left\{ \frac{202}{9} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 112x+168 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-112x} & = & \color{red}{112x} +168 \color{blue}{-112x} \color{blue}{-24} \\\Leftrightarrow & -91x& = & 144 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{144}{-91} \\\Leftrightarrow & x = \frac{-144}{91} & & \\ & V = \left\{ \frac{-144}{91} \right\} & \\\end{align}\)
7. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 9 }{ \color{blue}{21} })& = & (\frac{-35}{ \color{blue}{21} }x-\frac{126}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+9& = & -35x-126 \\\Leftrightarrow & 3x \color{red}{+9} \color{blue}{-9} \color{blue}{+35x} & = & \color{red}{-35x} -126 \color{blue}{+35x} \color{blue}{-9} \\\Leftrightarrow & 38x& = & -135 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{-135}{38} \\\Leftrightarrow & x = \frac{-135}{38} & & \\ & V = \left\{ \frac{-135}{38} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & -156x+1560 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{+156x} & = & \color{red}{-156x} +1560 \color{blue}{+156x} \color{blue}{-150} \\\Leftrightarrow & 221x& = & 1410 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{1410}{221} \\\Leftrightarrow & x = \frac{1410}{221} & & \\ & V = \left\{ \frac{1410}{221} \right\} & \\\end{align}\)
9. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x+\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & -99x+396 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{+99x} & = & \color{red}{-99x} +396 \color{blue}{+99x} \color{blue}{+48} \\\Leftrightarrow & 143x& = & 444 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{444}{143} \\\Leftrightarrow & x = \frac{444}{143} & & \\ & V = \left\{ \frac{444}{143} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{3}{5}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{48}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+25& = & 48x+400 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{-48x} & = & \color{red}{48x} +400 \color{blue}{-48x} \color{blue}{-25} \\\Leftrightarrow & -28x& = & 375 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{375}{-28} \\\Leftrightarrow & x = \frac{-375}{28} & & \\ & V = \left\{ \frac{-375}{28} \right\} & \\\end{align}\)
11. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{144}{ \color{blue}{120} }x+\frac{120}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 144x+120 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-144x} & = & \color{red}{144x} +120 \color{blue}{-144x} \color{blue}{+75} \\\Leftrightarrow & -124x& = & 195 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{195}{-124} \\\Leftrightarrow & x = \frac{-195}{124} & & \\ & V = \left\{ \frac{-195}{124} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & -220x+792 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{+220x} & = & \color{red}{-220x} +792 \color{blue}{+220x} \color{blue}{+48} \\\Leftrightarrow & 253x& = & 840 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{840}{253} \\\Leftrightarrow & x = \frac{840}{253} & & \\ & V = \left\{ \frac{840}{253} \right\} & \\\end{align}\)