Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 84 }{ \color{blue}{231} })& = & (\frac{308}{ \color{blue}{231} }x-\frac{462}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+84& = & 308x-462 \\\Leftrightarrow & 33x \color{red}{+84} \color{blue}{-84} \color{blue}{-308x} & = & \color{red}{308x} -462 \color{blue}{-308x} \color{blue}{-84} \\\Leftrightarrow & -275x& = & -546 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{-546}{-275} \\\Leftrightarrow & x = \frac{546}{275} & & \\ & V = \left\{ \frac{546}{275} \right\} & \\\end{align}\)
2. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{2}{3}x-6 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 84 }{ \color{blue}{273} })& = & (\frac{182}{ \color{blue}{273} }x-\frac{1638}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+84& = & 182x-1638 \\\Leftrightarrow & 39x \color{red}{+84} \color{blue}{-84} \color{blue}{-182x} & = & \color{red}{182x} -1638 \color{blue}{-182x} \color{blue}{-84} \\\Leftrightarrow & -143x& = & -1722 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-1722}{-143} \\\Leftrightarrow & x = \frac{1722}{143} & & \\ & V = \left\{ \frac{1722}{143} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x+\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & 40x+1920 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{-40x} & = & \color{red}{40x} +1920 \color{blue}{-40x} \color{blue}{+45} \\\Leftrightarrow & 8x& = & 1965 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{1965}{8} \\\Leftrightarrow & x = \frac{1965}{8} & & \\ & V = \left\{ \frac{1965}{8} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x+\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & 44x+396 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{-44x} & = & \color{red}{44x} +396 \color{blue}{-44x} \color{blue}{+48} \\\Leftrightarrow & -11x& = & 444 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{444}{-11} \\\Leftrightarrow & x = \frac{-444}{11} & & \\ & V = \left\{ \frac{-444}{11} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{4}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 24x+30 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-24x} & = & \color{red}{24x} +30 \color{blue}{-24x} \color{blue}{+8} \\\Leftrightarrow & -19x& = & 38 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{38}{-19} \\\Leftrightarrow & x = -2 & & \\ & V = \left\{ -2 \right\} & \\\end{align}\)
6. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{5}{4}x-4 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 21 }{ \color{blue}{112} })& = & (\frac{140}{ \color{blue}{112} }x-\frac{448}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-21& = & 140x-448 \\\Leftrightarrow & 16x \color{red}{-21} \color{blue}{+21} \color{blue}{-140x} & = & \color{red}{140x} -448 \color{blue}{-140x} \color{blue}{+21} \\\Leftrightarrow & -124x& = & -427 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{-427}{-124} \\\Leftrightarrow & x = \frac{427}{124} & & \\ & V = \left\{ \frac{427}{124} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 28 }{ \color{blue}{210} })& = & (\frac{105}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-28& = & 105x+1260 \\\Leftrightarrow & 30x \color{red}{-28} \color{blue}{+28} \color{blue}{-105x} & = & \color{red}{105x} +1260 \color{blue}{-105x} \color{blue}{+28} \\\Leftrightarrow & -75x& = & 1288 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{1288}{-75} \\\Leftrightarrow & x = \frac{-1288}{75} & & \\ & V = \left\{ \frac{-1288}{75} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 196x-336 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-196x} & = & \color{red}{196x} -336 \color{blue}{-196x} \color{blue}{+24} \\\Leftrightarrow & -175x& = & -312 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-312}{-175} \\\Leftrightarrow & x = \frac{312}{175} & & \\ & V = \left\{ \frac{312}{175} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{7}{3}x-3 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{182}{ \color{blue}{78} }x-\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-12& = & 182x-234 \\\Leftrightarrow & 39x \color{red}{-12} \color{blue}{+12} \color{blue}{-182x} & = & \color{red}{182x} -234 \color{blue}{-182x} \color{blue}{+12} \\\Leftrightarrow & -143x& = & -222 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-222}{-143} \\\Leftrightarrow & x = \frac{222}{143} & & \\ & V = \left\{ \frac{222}{143} \right\} & \\\end{align}\)
10. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 84 }{ \color{blue}{462} })& = & (\frac{77}{ \color{blue}{462} }x+\frac{2772}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-84& = & 77x+2772 \\\Leftrightarrow & 66x \color{red}{-84} \color{blue}{+84} \color{blue}{-77x} & = & \color{red}{77x} +2772 \color{blue}{-77x} \color{blue}{+84} \\\Leftrightarrow & -11x& = & 2856 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{2856}{-11} \\\Leftrightarrow & x = \frac{-2856}{11} & & \\ & V = \left\{ \frac{-2856}{11} \right\} & \\\end{align}\)
11. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 28 }{ \color{blue}{126} })& = & (\frac{21}{ \color{blue}{126} }x+\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-28& = & 21x+1008 \\\Leftrightarrow & 18x \color{red}{-28} \color{blue}{+28} \color{blue}{-21x} & = & \color{red}{21x} +1008 \color{blue}{-21x} \color{blue}{+28} \\\Leftrightarrow & -3x& = & 1036 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{1036}{-3} \\\Leftrightarrow & x = \frac{-1036}{3} & & \\ & V = \left\{ \frac{-1036}{3} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{14}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 15 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+15& = & 21x+84 \\\Leftrightarrow & 14x \color{red}{+15} \color{blue}{-15} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{-15} \\\Leftrightarrow & -7x& = & 69 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{69}{-7} \\\Leftrightarrow & x = \frac{-69}{7} & & \\ & V = \left\{ \frac{-69}{7} \right\} & \\\end{align}\)