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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 5x-90 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-5x} & = & \color{red}{5x} -90 \color{blue}{-5x} \color{blue}{+4} \\\Leftrightarrow & x& = & -86 \\ & V = \left\{ -86 \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{27}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 27x-36 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-27x} & = & \color{red}{27x} -36 \color{blue}{-27x} \color{blue}{+8} \\\Leftrightarrow & -21x& = & -28 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-28}{-21} \\\Leftrightarrow & x = \frac{4}{3} & & \\ & V = \left\{ \frac{4}{3} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & -176x+66 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+176x} & = & \color{red}{-176x} +66 \color{blue}{+176x} \color{blue}{+24} \\\Leftrightarrow & 209x& = & 90 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{90}{209} \\\Leftrightarrow & x = \frac{90}{209} & & \\ & V = \left\{ \frac{90}{209} \right\} & \\\end{align}\)
4. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x-\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & -168x-360 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{+168x} & = & \color{red}{-168x} -360 \color{blue}{+168x} \color{blue}{+75} \\\Leftrightarrow & 188x& = & -285 \\\Leftrightarrow & \frac{188x}{ \color{red}{188} }& = & \frac{-285}{188} \\\Leftrightarrow & x = \frac{-285}{188} & & \\ & V = \left\{ \frac{-285}{188} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-294 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -294 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -270 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-270}{-7} \\\Leftrightarrow & x = \frac{270}{7} & & \\ & V = \left\{ \frac{270}{7} \right\} & \\\end{align}\)
6. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{5}{2}x+6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 3 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x+\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-3& = & 35x+84 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-35x} & = & \color{red}{35x} +84 \color{blue}{-35x} \color{blue}{+3} \\\Leftrightarrow & -33x& = & 87 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{87}{-33} \\\Leftrightarrow & x = \frac{-29}{11} & & \\ & V = \left\{ \frac{-29}{11} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{1}{3}x-2 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & 44x-264 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-44x} & = & \color{red}{44x} -264 \color{blue}{-44x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & -288 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-288}{-11} \\\Leftrightarrow & x = \frac{288}{11} & & \\ & V = \left\{ \frac{288}{11} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 112x+588 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-112x} & = & \color{red}{112x} +588 \color{blue}{-112x} \color{blue}{+24} \\\Leftrightarrow & -91x& = & 612 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{612}{-91} \\\Leftrightarrow & x = \frac{-612}{91} & & \\ & V = \left\{ \frac{-612}{91} \right\} & \\\end{align}\)
9. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 40 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x+\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-40& = & 65x+520 \\\Leftrightarrow & 52x \color{red}{-40} \color{blue}{+40} \color{blue}{-65x} & = & \color{red}{65x} +520 \color{blue}{-65x} \color{blue}{+40} \\\Leftrightarrow & -13x& = & 560 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{560}{-13} \\\Leftrightarrow & x = \frac{-560}{13} & & \\ & V = \left\{ \frac{-560}{13} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 196x+84 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-196x} & = & \color{red}{196x} +84 \color{blue}{-196x} \color{blue}{+36} \\\Leftrightarrow & -175x& = & 120 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{120}{-175} \\\Leftrightarrow & x = \frac{-24}{35} & & \\ & V = \left\{ \frac{-24}{35} \right\} & \\\end{align}\)
11. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{4}x-4 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 84 }{ \color{blue}{364} })& = & (\frac{91}{ \color{blue}{364} }x-\frac{1456}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-84& = & 91x-1456 \\\Leftrightarrow & 52x \color{red}{-84} \color{blue}{+84} \color{blue}{-91x} & = & \color{red}{91x} -1456 \color{blue}{-91x} \color{blue}{+84} \\\Leftrightarrow & -39x& = & -1372 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-1372}{-39} \\\Leftrightarrow & x = \frac{1372}{39} & & \\ & V = \left\{ \frac{1372}{39} \right\} & \\\end{align}\)
12. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{7}{3}x-3 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{28}{ \color{blue}{12} }x-\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & 28x-36 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{-28x} & = & \color{red}{28x} -36 \color{blue}{-28x} \color{blue}{+10} \\\Leftrightarrow & -25x& = & -26 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-26}{-25} \\\Leftrightarrow & x = \frac{26}{25} & & \\ & V = \left\{ \frac{26}{25} \right\} & \\\end{align}\)