Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{8}& = & \frac{7}{4}x+4 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }- \frac{ 15 }{ \color{blue}{40} })& = & (\frac{70}{ \color{blue}{40} }x+\frac{160}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x-15& = & 70x+160 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-70x} & = & \color{red}{70x} +160 \color{blue}{-70x} \color{blue}{+15} \\\Leftrightarrow & -62x& = & 175 \\\Leftrightarrow & \frac{-62x}{ \color{red}{-62} }& = & \frac{175}{-62} \\\Leftrightarrow & x = \frac{-175}{62} & & \\ & V = \left\{ \frac{-175}{62} \right\} & \\\end{align}\)
2. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 56 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{770}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+56& = & 77x-770 \\\Leftrightarrow & 22x \color{red}{+56} \color{blue}{-56} \color{blue}{-77x} & = & \color{red}{77x} -770 \color{blue}{-77x} \color{blue}{-56} \\\Leftrightarrow & -55x& = & -826 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-826}{-55} \\\Leftrightarrow & x = \frac{826}{55} & & \\ & V = \left\{ \frac{826}{55} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 168x-420 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-168x} & = & \color{red}{168x} -420 \color{blue}{-168x} \color{blue}{-80} \\\Leftrightarrow & -133x& = & -500 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-500}{-133} \\\Leftrightarrow & x = \frac{500}{133} & & \\ & V = \left\{ \frac{500}{133} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & -462x-2640 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+462x} & = & \color{red}{-462x} -2640 \color{blue}{+462x} \color{blue}{+60} \\\Leftrightarrow & 517x& = & -2580 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{-2580}{517} \\\Leftrightarrow & x = \frac{-2580}{517} & & \\ & V = \left\{ \frac{-2580}{517} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -84x-1260 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+84x} & = & \color{red}{-84x} -1260 \color{blue}{+84x} \color{blue}{-90} \\\Leftrightarrow & 119x& = & -1350 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-1350}{119} \\\Leftrightarrow & x = \frac{-1350}{119} & & \\ & V = \left\{ \frac{-1350}{119} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 63x-84 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-63x} & = & \color{red}{63x} -84 \color{blue}{-63x} \color{blue}{-12} \\\Leftrightarrow & -49x& = & -96 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-96}{-49} \\\Leftrightarrow & x = \frac{96}{49} & & \\ & V = \left\{ \frac{96}{49} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{-8}{3}x+3 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x+\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & -176x+198 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+176x} & = & \color{red}{-176x} +198 \color{blue}{+176x} \color{blue}{+24} \\\Leftrightarrow & 209x& = & 222 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{222}{209} \\\Leftrightarrow & x = \frac{222}{209} & & \\ & V = \left\{ \frac{222}{209} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{14}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 25 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-25& = & 35x-280 \\\Leftrightarrow & 14x \color{red}{-25} \color{blue}{+25} \color{blue}{-35x} & = & \color{red}{35x} -280 \color{blue}{-35x} \color{blue}{+25} \\\Leftrightarrow & -21x& = & -255 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-255}{-21} \\\Leftrightarrow & x = \frac{85}{7} & & \\ & V = \left\{ \frac{85}{7} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -112x-252 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+112x} & = & \color{red}{-112x} -252 \color{blue}{+112x} \color{blue}{+24} \\\Leftrightarrow & 133x& = & -228 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-228}{133} \\\Leftrightarrow & x = \frac{-12}{7} & & \\ & V = \left\{ \frac{-12}{7} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-60& = & 55x+990 \\\Leftrightarrow & 66x \color{red}{-60} \color{blue}{+60} \color{blue}{-55x} & = & \color{red}{55x} +990 \color{blue}{-55x} \color{blue}{+60} \\\Leftrightarrow & 11x& = & 1050 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{1050}{11} \\\Leftrightarrow & x = \frac{1050}{11} & & \\ & V = \left\{ \frac{1050}{11} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 14x-252 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-14x} & = & \color{red}{14x} -252 \color{blue}{-14x} \color{blue}{-12} \\\Leftrightarrow & 7x& = & -264 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-264}{7} \\\Leftrightarrow & x = \frac{-264}{7} & & \\ & V = \left\{ \frac{-264}{7} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -12x-60 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+12x} & = & \color{red}{-12x} -60 \color{blue}{+12x} \color{blue}{+9} \\\Leftrightarrow & 17x& = & -51 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-51}{17} \\\Leftrightarrow & x = -3 & & \\ & V = \left\{ -3 \right\} & \\\end{align}\)