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

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

Verbetersleutel

1. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 9 }{ \color{blue}{21} })& = & (\frac{49}{ \color{blue}{21} }x+\frac{105}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-9& = & 49x+105 \\\Leftrightarrow & 3x \color{red}{-9} \color{blue}{+9} \color{blue}{-49x} & = & \color{red}{49x} +105 \color{blue}{-49x} \color{blue}{+9} \\\Leftrightarrow & -46x& = & 114 \\\Leftrightarrow & \frac{-46x}{ \color{red}{-46} }& = & \frac{114}{-46} \\\Leftrightarrow & x = \frac{-57}{23} & & \\ & V = \left\{ \frac{-57}{23} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{4}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+24& = & 21x-168 \\\Leftrightarrow & 28x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & 7x& = & -192 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-192}{7} \\\Leftrightarrow & x = \frac{-192}{7} & & \\ & V = \left\{ \frac{-192}{7} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & 35x-420 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{-35x} & = & \color{red}{35x} -420 \color{blue}{-35x} \color{blue}{+21} \\\Leftrightarrow & -25x& = & -399 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-399}{-25} \\\Leftrightarrow & x = \frac{399}{25} & & \\ & V = \left\{ \frac{399}{25} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+24& = & 39x-546 \\\Leftrightarrow & 26x \color{red}{+24} \color{blue}{-24} \color{blue}{-39x} & = & \color{red}{39x} -546 \color{blue}{-39x} \color{blue}{-24} \\\Leftrightarrow & -13x& = & -570 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-570}{-13} \\\Leftrightarrow & x = \frac{570}{13} & & \\ & V = \left\{ \frac{570}{13} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{3}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & 126x-1050 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{-126x} & = & \color{red}{126x} -1050 \color{blue}{-126x} \color{blue}{-150} \\\Leftrightarrow & -91x& = & -1200 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-1200}{-91} \\\Leftrightarrow & x = \frac{1200}{91} & & \\ & V = \left\{ \frac{1200}{91} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & -50x+90 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+50x} & = & \color{red}{-50x} +90 \color{blue}{+50x} \color{blue}{-8} \\\Leftrightarrow & 65x& = & 82 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{82}{65} \\\Leftrightarrow & x = \frac{82}{65} & & \\ & V = \left\{ \frac{82}{65} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-4}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -168x-840 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+168x} & = & \color{red}{-168x} -840 \color{blue}{+168x} \color{blue}{+90} \\\Leftrightarrow & 203x& = & -750 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-750}{203} \\\Leftrightarrow & x = \frac{-750}{203} & & \\ & V = \left\{ \frac{-750}{203} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & -140x-672 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{+140x} & = & \color{red}{-140x} -672 \color{blue}{+140x} \color{blue}{-36} \\\Leftrightarrow & 161x& = & -708 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-708}{161} \\\Leftrightarrow & x = \frac{-708}{161} & & \\ & V = \left\{ \frac{-708}{161} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 15x-30 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-15x} & = & \color{red}{15x} -30 \color{blue}{-15x} \color{blue}{+9} \\\Leftrightarrow & -5x& = & -21 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-21}{-5} \\\Leftrightarrow & x = \frac{21}{5} & & \\ & V = \left\{ \frac{21}{5} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{3}{4}x-6 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{45}{ \color{blue}{60} }x-\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-8& = & 45x-360 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-45x} & = & \color{red}{45x} -360 \color{blue}{-45x} \color{blue}{+8} \\\Leftrightarrow & -33x& = & -352 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-352}{-33} \\\Leftrightarrow & x = \frac{32}{3} & & \\ & V = \left\{ \frac{32}{3} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x-\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & -30x-72 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{+30x} & = & \color{red}{-30x} -72 \color{blue}{+30x} \color{blue}{-4} \\\Leftrightarrow & 39x& = & -76 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-76}{39} \\\Leftrightarrow & x = \frac{-76}{39} & & \\ & V = \left\{ \frac{-76}{39} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & -84x+60 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{+84x} & = & \color{red}{-84x} +60 \color{blue}{+84x} \color{blue}{-25} \\\Leftrightarrow & 94x& = & 35 \\\Leftrightarrow & \frac{94x}{ \color{red}{94} }& = & \frac{35}{94} \\\Leftrightarrow & x = \frac{35}{94} & & \\ & V = \left\{ \frac{35}{94} \right\} & \\\end{align}\)