Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-168 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -144 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-144}{-7} \\\Leftrightarrow & x = \frac{144}{7} & & \\ & V = \left\{ \frac{144}{7} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{8}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+15& = & 12x-96 \\\Leftrightarrow & 8x \color{red}{+15} \color{blue}{-15} \color{blue}{-12x} & = & \color{red}{12x} -96 \color{blue}{-12x} \color{blue}{-15} \\\Leftrightarrow & -4x& = & -111 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-111}{-4} \\\Leftrightarrow & x = \frac{111}{4} & & \\ & V = \left\{ \frac{111}{4} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 7, 10 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{-7}{4}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{20x}{ \color{blue}{140} }- \frac{ 42 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 20x-42& = & -245x-980 \\\Leftrightarrow & 20x \color{red}{-42} \color{blue}{+42} \color{blue}{+245x} & = & \color{red}{-245x} -980 \color{blue}{+245x} \color{blue}{+42} \\\Leftrightarrow & 265x& = & -938 \\\Leftrightarrow & \frac{265x}{ \color{red}{265} }& = & \frac{-938}{265} \\\Leftrightarrow & x = \frac{-938}{265} & & \\ & V = \left\{ \frac{-938}{265} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{3}{5}x+4 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{84}{ \color{blue}{140} }x+\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 84x+560 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-84x} & = & \color{red}{84x} +560 \color{blue}{-84x} \color{blue}{-80} \\\Leftrightarrow & -49x& = & 480 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{480}{-49} \\\Leftrightarrow & x = \frac{-480}{49} & & \\ & V = \left\{ \frac{-480}{49} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{7}{6}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{49}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+18& = & 49x-126 \\\Leftrightarrow & 6x \color{red}{+18} \color{blue}{-18} \color{blue}{-49x} & = & \color{red}{49x} -126 \color{blue}{-49x} \color{blue}{-18} \\\Leftrightarrow & -43x& = & -144 \\\Leftrightarrow & \frac{-43x}{ \color{red}{-43} }& = & \frac{-144}{-43} \\\Leftrightarrow & x = \frac{144}{43} & & \\ & V = \left\{ \frac{144}{43} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -462x-2640 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+462x} & = & \color{red}{-462x} -2640 \color{blue}{+462x} \color{blue}{+150} \\\Leftrightarrow & 517x& = & -2490 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{-2490}{517} \\\Leftrightarrow & x = \frac{-2490}{517} & & \\ & V = \left\{ \frac{-2490}{517} \right\} & \\\end{align}\)
7. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{8}{ \color{blue}{24} }x+\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+15& = & 8x+168 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{-8x} & = & \color{red}{8x} +168 \color{blue}{-8x} \color{blue}{-15} \\\Leftrightarrow & -2x& = & 153 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{153}{-2} \\\Leftrightarrow & x = \frac{-153}{2} & & \\ & V = \left\{ \frac{-153}{2} \right\} & \\\end{align}\)
8. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{55}{ \color{blue}{165} }x+\frac{660}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+75& = & 55x+660 \\\Leftrightarrow & 33x \color{red}{+75} \color{blue}{-75} \color{blue}{-55x} & = & \color{red}{55x} +660 \color{blue}{-55x} \color{blue}{-75} \\\Leftrightarrow & -22x& = & 585 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{585}{-22} \\\Leftrightarrow & x = \frac{-585}{22} & & \\ & V = \left\{ \frac{-585}{22} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 196x-672 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-196x} & = & \color{red}{196x} -672 \color{blue}{-196x} \color{blue}{+24} \\\Leftrightarrow & -175x& = & -648 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-648}{-175} \\\Leftrightarrow & x = \frac{648}{175} & & \\ & V = \left\{ \frac{648}{175} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -294x+840 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+294x} & = & \color{red}{-294x} +840 \color{blue}{+294x} \color{blue}{+90} \\\Leftrightarrow & 329x& = & 930 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{930}{329} \\\Leftrightarrow & x = \frac{930}{329} & & \\ & V = \left\{ \frac{930}{329} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{4}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & 24x+210 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{-24x} & = & \color{red}{24x} +210 \color{blue}{-24x} \color{blue}{-8} \\\Leftrightarrow & -19x& = & 202 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{202}{-19} \\\Leftrightarrow & x = \frac{-202}{19} & & \\ & V = \left\{ \frac{-202}{19} \right\} & \\\end{align}\)
12. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{2}{3}x-5 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 63 }{ \color{blue}{336} })& = & (\frac{224}{ \color{blue}{336} }x-\frac{1680}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-63& = & 224x-1680 \\\Leftrightarrow & 48x \color{red}{-63} \color{blue}{+63} \color{blue}{-224x} & = & \color{red}{224x} -1680 \color{blue}{-224x} \color{blue}{+63} \\\Leftrightarrow & -176x& = & -1617 \\\Leftrightarrow & \frac{-176x}{ \color{red}{-176} }& = & \frac{-1617}{-176} \\\Leftrightarrow & x = \frac{147}{16} & & \\ & V = \left\{ \frac{147}{16} \right\} & \\\end{align}\)