Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{315 is het kleinste gemene veelvoud van 7, 9 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{315.} (\frac{45x}{ \color{blue}{315} }- \frac{ 175 }{ \color{blue}{315} })& = & (\frac{-441}{ \color{blue}{315} }x+\frac{315}{ \color{blue}{315} }) \color{blue}{.315} \\\Leftrightarrow & 45x-175& = & -441x+315 \\\Leftrightarrow & 45x \color{red}{-175} \color{blue}{+175} \color{blue}{+441x} & = & \color{red}{-441x} +315 \color{blue}{+441x} \color{blue}{+175} \\\Leftrightarrow & 486x& = & 490 \\\Leftrightarrow & \frac{486x}{ \color{red}{486} }& = & \frac{490}{486} \\\Leftrightarrow & x = \frac{245}{243} & & \\ & V = \left\{ \frac{245}{243} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 252x-420 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-252x} & = & \color{red}{252x} -420 \color{blue}{-252x} \color{blue}{-120} \\\Leftrightarrow & -217x& = & -540 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-540}{-217} \\\Leftrightarrow & x = \frac{540}{217} & & \\ & V = \left\{ \frac{540}{217} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{14}& = & \frac{5}{2}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 15 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-15& = & 105x+210 \\\Leftrightarrow & 14x \color{red}{-15} \color{blue}{+15} \color{blue}{-105x} & = & \color{red}{105x} +210 \color{blue}{-105x} \color{blue}{+15} \\\Leftrightarrow & -91x& = & 225 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{225}{-91} \\\Leftrightarrow & x = \frac{-225}{91} & & \\ & V = \left\{ \frac{-225}{91} \right\} & \\\end{align}\)
4. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 10 }{ \color{blue}{35} })& = & (\frac{21}{ \color{blue}{35} }x+\frac{245}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-10& = & 21x+245 \\\Leftrightarrow & 5x \color{red}{-10} \color{blue}{+10} \color{blue}{-21x} & = & \color{red}{21x} +245 \color{blue}{-21x} \color{blue}{+10} \\\Leftrightarrow & -16x& = & 255 \\\Leftrightarrow & \frac{-16x}{ \color{red}{-16} }& = & \frac{255}{-16} \\\Leftrightarrow & x = \frac{-255}{16} & & \\ & V = \left\{ \frac{-255}{16} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x-\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+60& = & -70x-735 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{+70x} & = & \color{red}{-70x} -735 \color{blue}{+70x} \color{blue}{-60} \\\Leftrightarrow & 91x& = & -795 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-795}{91} \\\Leftrightarrow & x = \frac{-795}{91} & & \\ & V = \left\{ \frac{-795}{91} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -24x+90 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+24x} & = & \color{red}{-24x} +90 \color{blue}{+24x} \color{blue}{+4} \\\Leftrightarrow & 29x& = & 94 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{94}{29} \\\Leftrightarrow & x = \frac{94}{29} & & \\ & V = \left\{ \frac{94}{29} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & -84x-420 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{+84x} & = & \color{red}{-84x} -420 \color{blue}{+84x} \color{blue}{+45} \\\Leftrightarrow & 119x& = & -375 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-375}{119} \\\Leftrightarrow & x = \frac{-375}{119} & & \\ & V = \left\{ \frac{-375}{119} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-30& = & 35x-350 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-35x} & = & \color{red}{35x} -350 \color{blue}{-35x} \color{blue}{+30} \\\Leftrightarrow & -21x& = & -320 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-320}{-21} \\\Leftrightarrow & x = \frac{320}{21} & & \\ & V = \left\{ \frac{320}{21} \right\} & \\\end{align}\)
9. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{-66}{ \color{blue}{165} }x+\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+75& = & -66x+330 \\\Leftrightarrow & 55x \color{red}{+75} \color{blue}{-75} \color{blue}{+66x} & = & \color{red}{-66x} +330 \color{blue}{+66x} \color{blue}{-75} \\\Leftrightarrow & 121x& = & 255 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{255}{121} \\\Leftrightarrow & x = \frac{255}{121} & & \\ & V = \left\{ \frac{255}{121} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 15 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+15& = & 98x-168 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{-98x} & = & \color{red}{98x} -168 \color{blue}{-98x} \color{blue}{-15} \\\Leftrightarrow & -92x& = & -183 \\\Leftrightarrow & \frac{-92x}{ \color{red}{-92} }& = & \frac{-183}{-92} \\\Leftrightarrow & x = \frac{183}{92} & & \\ & V = \left\{ \frac{183}{92} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{14}& = & \frac{3}{5}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 15 }{ \color{blue}{70} })& = & (\frac{42}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+15& = & 42x-280 \\\Leftrightarrow & 35x \color{red}{+15} \color{blue}{-15} \color{blue}{-42x} & = & \color{red}{42x} -280 \color{blue}{-42x} \color{blue}{-15} \\\Leftrightarrow & -7x& = & -295 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-295}{-7} \\\Leftrightarrow & x = \frac{295}{7} & & \\ & V = \left\{ \frac{295}{7} \right\} & \\\end{align}\)
12. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{12}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{6}{ \color{blue}{12} }x-\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-5& = & 6x-84 \\\Leftrightarrow & 4x \color{red}{-5} \color{blue}{+5} \color{blue}{-6x} & = & \color{red}{6x} -84 \color{blue}{-6x} \color{blue}{+5} \\\Leftrightarrow & -2x& = & -79 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-79}{-2} \\\Leftrightarrow & x = \frac{79}{2} & & \\ & V = \left\{ \frac{79}{2} \right\} & \\\end{align}\)