Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{3}{4}x-4 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{27}{ \color{blue}{36} }x-\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & 27x-144 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-27x} & = & \color{red}{27x} -144 \color{blue}{-27x} \color{blue}{+8} \\\Leftrightarrow & -15x& = & -136 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-136}{-15} \\\Leftrightarrow & x = \frac{136}{15} & & \\ & V = \left\{ \frac{136}{15} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x+\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & -88x+924 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{+88x} & = & \color{red}{-88x} +924 \color{blue}{+88x} \color{blue}{-48} \\\Leftrightarrow & 121x& = & 876 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{876}{121} \\\Leftrightarrow & x = \frac{876}{121} & & \\ & V = \left\{ \frac{876}{121} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 42x+420 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-42x} & = & \color{red}{42x} +420 \color{blue}{-42x} \color{blue}{-90} \\\Leftrightarrow & -7x& = & 330 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{330}{-7} \\\Leftrightarrow & x = \frac{-330}{7} & & \\ & V = \left\{ \frac{-330}{7} \right\} & \\\end{align}\)
4. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 84 }{ \color{blue}{231} })& = & (\frac{539}{ \color{blue}{231} }x+\frac{693}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-84& = & 539x+693 \\\Leftrightarrow & 33x \color{red}{-84} \color{blue}{+84} \color{blue}{-539x} & = & \color{red}{539x} +693 \color{blue}{-539x} \color{blue}{+84} \\\Leftrightarrow & -506x& = & 777 \\\Leftrightarrow & \frac{-506x}{ \color{red}{-506} }& = & \frac{777}{-506} \\\Leftrightarrow & x = \frac{-777}{506} & & \\ & V = \left\{ \frac{-777}{506} \right\} & \\\end{align}\)
5. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-100}{ \color{blue}{120} }x+\frac{720}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x-75& = & -100x+720 \\\Leftrightarrow & 24x \color{red}{-75} \color{blue}{+75} \color{blue}{+100x} & = & \color{red}{-100x} +720 \color{blue}{+100x} \color{blue}{+75} \\\Leftrightarrow & 124x& = & 795 \\\Leftrightarrow & \frac{124x}{ \color{red}{124} }& = & \frac{795}{124} \\\Leftrightarrow & x = \frac{795}{124} & & \\ & V = \left\{ \frac{795}{124} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-5}{6}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+24& = & -35x+42 \\\Leftrightarrow & 6x \color{red}{+24} \color{blue}{-24} \color{blue}{+35x} & = & \color{red}{-35x} +42 \color{blue}{+35x} \color{blue}{-24} \\\Leftrightarrow & 41x& = & 18 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{18}{41} \\\Leftrightarrow & x = \frac{18}{41} & & \\ & V = \left\{ \frac{18}{41} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x-\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 21x-84 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-21x} & = & \color{red}{21x} -84 \color{blue}{-21x} \color{blue}{-4} \\\Leftrightarrow & -19x& = & -88 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-88}{-19} \\\Leftrightarrow & x = \frac{88}{19} & & \\ & V = \left\{ \frac{88}{19} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x+\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-60& = & -196x+280 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+196x} & = & \color{red}{-196x} +280 \color{blue}{+196x} \color{blue}{+60} \\\Leftrightarrow & 231x& = & 340 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{340}{231} \\\Leftrightarrow & x = \frac{340}{231} & & \\ & V = \left\{ \frac{340}{231} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x-\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & 154x-462 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{-154x} & = & \color{red}{154x} -462 \color{blue}{-154x} \color{blue}{-30} \\\Leftrightarrow & -121x& = & -492 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-492}{-121} \\\Leftrightarrow & x = \frac{492}{121} & & \\ & V = \left\{ \frac{492}{121} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+8& = & 15x+90 \\\Leftrightarrow & 10x \color{red}{+8} \color{blue}{-8} \color{blue}{-15x} & = & \color{red}{15x} +90 \color{blue}{-15x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & 82 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{82}{-5} \\\Leftrightarrow & x = \frac{-82}{5} & & \\ & V = \left\{ \frac{-82}{5} \right\} & \\\end{align}\)
11. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{9}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 20 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x-\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+20& = & -60x-180 \\\Leftrightarrow & 9x \color{red}{+20} \color{blue}{-20} \color{blue}{+60x} & = & \color{red}{-60x} -180 \color{blue}{+60x} \color{blue}{-20} \\\Leftrightarrow & 69x& = & -200 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{-200}{69} \\\Leftrightarrow & x = \frac{-200}{69} & & \\ & V = \left\{ \frac{-200}{69} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{2}{5}x+4 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{52}{ \color{blue}{130} }x+\frac{520}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & 52x+520 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{-52x} & = & \color{red}{52x} +520 \color{blue}{-52x} \color{blue}{-20} \\\Leftrightarrow & 13x& = & 500 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{500}{13} \\\Leftrightarrow & x = \frac{500}{13} & & \\ & V = \left\{ \frac{500}{13} \right\} & \\\end{align}\)