Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x-\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 6x-72 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} -72 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & 3x& = & -76 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-76}{3} \\\Leftrightarrow & x = \frac{-76}{3} & & \\ & V = \left\{ \frac{-76}{3} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{26}{ \color{blue}{130} }x+\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & 26x+260 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{-26x} & = & \color{red}{26x} +260 \color{blue}{-26x} \color{blue}{-20} \\\Leftrightarrow & 39x& = & 240 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{240}{39} \\\Leftrightarrow & x = \frac{80}{13} & & \\ & V = \left\{ \frac{80}{13} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-56}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+60& = & -56x-420 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+56x} & = & \color{red}{-56x} -420 \color{blue}{+56x} \color{blue}{-60} \\\Leftrightarrow & 91x& = & -480 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-480}{91} \\\Leftrightarrow & x = \frac{-480}{91} & & \\ & V = \left\{ \frac{-480}{91} \right\} & \\\end{align}\)
4. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 70 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{182}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-70& = & 91x-182 \\\Leftrightarrow & 26x \color{red}{-70} \color{blue}{+70} \color{blue}{-91x} & = & \color{red}{91x} -182 \color{blue}{-91x} \color{blue}{+70} \\\Leftrightarrow & -65x& = & -112 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-112}{-65} \\\Leftrightarrow & x = \frac{112}{65} & & \\ & V = \left\{ \frac{112}{65} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & 196x+168 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-196x} & = & \color{red}{196x} +168 \color{blue}{-196x} \color{blue}{+18} \\\Leftrightarrow & -175x& = & 186 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{186}{-175} \\\Leftrightarrow & x = \frac{-186}{175} & & \\ & V = \left\{ \frac{-186}{175} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x+42 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} +42 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 18 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{18}{-7} \\\Leftrightarrow & x = \frac{-18}{7} & & \\ & V = \left\{ \frac{-18}{7} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{7}{2}x+8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x+\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 49x+112 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-49x} & = & \color{red}{49x} +112 \color{blue}{-49x} \color{blue}{+4} \\\Leftrightarrow & -47x& = & 116 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{116}{-47} \\\Leftrightarrow & x = \frac{-116}{47} & & \\ & V = \left\{ \frac{-116}{47} \right\} & \\\end{align}\)
8. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x-\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & 40x-1920 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{-40x} & = & \color{red}{40x} -1920 \color{blue}{-40x} \color{blue}{+45} \\\Leftrightarrow & 8x& = & -1875 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-1875}{8} \\\Leftrightarrow & x = \frac{-1875}{8} & & \\ & V = \left\{ \frac{-1875}{8} \right\} & \\\end{align}\)
9. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{4}{5}x+3 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 60 }{ \color{blue}{165} })& = & (\frac{132}{ \color{blue}{165} }x+\frac{495}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+60& = & 132x+495 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-132x} & = & \color{red}{132x} +495 \color{blue}{-132x} \color{blue}{-60} \\\Leftrightarrow & -77x& = & 435 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{435}{-77} \\\Leftrightarrow & x = \frac{-435}{77} & & \\ & V = \left\{ \frac{-435}{77} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{-7}{4}x-6 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-63}{ \color{blue}{36} }x-\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & -63x-216 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{+63x} & = & \color{red}{-63x} -216 \color{blue}{+63x} \color{blue}{-8} \\\Leftrightarrow & 75x& = & -224 \\\Leftrightarrow & \frac{75x}{ \color{red}{75} }& = & \frac{-224}{75} \\\Leftrightarrow & x = \frac{-224}{75} & & \\ & V = \left\{ \frac{-224}{75} \right\} & \\\end{align}\)
11. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }- \frac{ 112 }{ \color{blue}{420} })& = & (\frac{-315}{ \color{blue}{420} }x-\frac{2940}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x-112& = & -315x-2940 \\\Leftrightarrow & 60x \color{red}{-112} \color{blue}{+112} \color{blue}{+315x} & = & \color{red}{-315x} -2940 \color{blue}{+315x} \color{blue}{+112} \\\Leftrightarrow & 375x& = & -2828 \\\Leftrightarrow & \frac{375x}{ \color{red}{375} }& = & \frac{-2828}{375} \\\Leftrightarrow & x = \frac{-2828}{375} & & \\ & V = \left\{ \frac{-2828}{375} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 100 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+100& = & -196x-560 \\\Leftrightarrow & 35x \color{red}{+100} \color{blue}{-100} \color{blue}{+196x} & = & \color{red}{-196x} -560 \color{blue}{+196x} \color{blue}{-100} \\\Leftrightarrow & 231x& = & -660 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-660}{231} \\\Leftrightarrow & x = \frac{-20}{7} & & \\ & V = \left\{ \frac{-20}{7} \right\} & \\\end{align}\)