Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & 468x+3120 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{-468x} & = & \color{red}{468x} +3120 \color{blue}{-468x} \color{blue}{-150} \\\Leftrightarrow & -403x& = & 2970 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{2970}{-403} \\\Leftrightarrow & x = \frac{-2970}{403} & & \\ & V = \left\{ \frac{-2970}{403} \right\} & \\\end{align}\)
2. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{9}& = & \frac{-3}{4}x-8 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 20 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x-\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+20& = & -27x-288 \\\Leftrightarrow & 12x \color{red}{+20} \color{blue}{-20} \color{blue}{+27x} & = & \color{red}{-27x} -288 \color{blue}{+27x} \color{blue}{-20} \\\Leftrightarrow & 39x& = & -308 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-308}{39} \\\Leftrightarrow & x = \frac{-308}{39} & & \\ & V = \left\{ \frac{-308}{39} \right\} & \\\end{align}\)
3. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 6 }{ \color{blue}{21} })& = & (\frac{28}{ \color{blue}{21} }x+\frac{105}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-6& = & 28x+105 \\\Leftrightarrow & 3x \color{red}{-6} \color{blue}{+6} \color{blue}{-28x} & = & \color{red}{28x} +105 \color{blue}{-28x} \color{blue}{+6} \\\Leftrightarrow & -25x& = & 111 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{111}{-25} \\\Leftrightarrow & x = \frac{-111}{25} & & \\ & V = \left\{ \frac{-111}{25} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 196x+420 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-196x} & = & \color{red}{196x} +420 \color{blue}{-196x} \color{blue}{-36} \\\Leftrightarrow & -175x& = & 384 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{384}{-175} \\\Leftrightarrow & x = \frac{-384}{175} & & \\ & V = \left\{ \frac{-384}{175} \right\} & \\\end{align}\)
5. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{5}{3}x-8 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{30}{ \color{blue}{18} }x-\frac{144}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & 30x-144 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-30x} & = & \color{red}{30x} -144 \color{blue}{-30x} \color{blue}{+8} \\\Leftrightarrow & -21x& = & -136 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-136}{-21} \\\Leftrightarrow & x = \frac{136}{21} & & \\ & V = \left\{ \frac{136}{21} \right\} & \\\end{align}\)
6. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 84 }{ \color{blue}{273} })& = & (\frac{91}{ \color{blue}{273} }x+\frac{1365}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-84& = & 91x+1365 \\\Leftrightarrow & 39x \color{red}{-84} \color{blue}{+84} \color{blue}{-91x} & = & \color{red}{91x} +1365 \color{blue}{-91x} \color{blue}{+84} \\\Leftrightarrow & -52x& = & 1449 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{1449}{-52} \\\Leftrightarrow & x = \frac{-1449}{52} & & \\ & V = \left\{ \frac{-1449}{52} \right\} & \\\end{align}\)
7. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 12 }{ \color{blue}{21} })& = & (\frac{49}{ \color{blue}{21} }x+\frac{21}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-12& = & 49x+21 \\\Leftrightarrow & 3x \color{red}{-12} \color{blue}{+12} \color{blue}{-49x} & = & \color{red}{49x} +21 \color{blue}{-49x} \color{blue}{+12} \\\Leftrightarrow & -46x& = & 33 \\\Leftrightarrow & \frac{-46x}{ \color{red}{-46} }& = & \frac{33}{-46} \\\Leftrightarrow & x = \frac{-33}{46} & & \\ & V = \left\{ \frac{-33}{46} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-18& = & -45x+60 \\\Leftrightarrow & 20x \color{red}{-18} \color{blue}{+18} \color{blue}{+45x} & = & \color{red}{-45x} +60 \color{blue}{+45x} \color{blue}{+18} \\\Leftrightarrow & 65x& = & 78 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{78}{65} \\\Leftrightarrow & x = \frac{6}{5} & & \\ & V = \left\{ \frac{6}{5} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 6x-180 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} -180 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & -x& = & -184 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-184}{-1} \\\Leftrightarrow & x = 184 & & \\ & V = \left\{ 184 \right\} & \\\end{align}\)
10. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x+\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & -192x+1440 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{+192x} & = & \color{red}{-192x} +1440 \color{blue}{+192x} \color{blue}{+45} \\\Leftrightarrow & 232x& = & 1485 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{1485}{232} \\\Leftrightarrow & x = \frac{1485}{232} & & \\ & V = \left\{ \frac{1485}{232} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{5}{2}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{225}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 225x+540 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-225x} & = & \color{red}{225x} +540 \color{blue}{-225x} \color{blue}{+40} \\\Leftrightarrow & -207x& = & 580 \\\Leftrightarrow & \frac{-207x}{ \color{red}{-207} }& = & \frac{580}{-207} \\\Leftrightarrow & x = \frac{-580}{207} & & \\ & V = \left\{ \frac{-580}{207} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{7}{6}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{35}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 35x-90 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-35x} & = & \color{red}{35x} -90 \color{blue}{-35x} \color{blue}{+8} \\\Leftrightarrow & -29x& = & -82 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-82}{-29} \\\Leftrightarrow & x = \frac{82}{29} & & \\ & V = \left\{ \frac{82}{29} \right\} & \\\end{align}\)