Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{5}{13}=\frac{-7}{5}x+2\)
2. \(\frac{x}{2}+\frac{4}{7}=\frac{-5}{3}x-2\)
3. \(\frac{x}{3}+\frac{3}{7}=\frac{1}{4}x+6\)
4. \(\frac{x}{4}+\frac{4}{7}=\frac{-2}{5}x+5\)
5. \(\frac{x}{7}-\frac{3}{14}=\frac{7}{4}x-5\)
6. \(\frac{x}{2}-\frac{2}{7}=\frac{-5}{3}x-5\)
7. \(\frac{x}{6}+\frac{3}{7}=\frac{-7}{5}x+8\)
8. \(\frac{x}{5}+\frac{2}{7}=\frac{5}{2}x+7\)
9. \(\frac{x}{3}+\frac{3}{11}=\frac{1}{2}x-6\)
10. \(\frac{x}{6}+\frac{3}{8}=\frac{-7}{5}x-5\)
11. \(\frac{x}{6}+\frac{5}{7}=\frac{-7}{5}x+2\)
12. \(\frac{x}{6}+\frac{3}{10}=\frac{-4}{5}x+4\)

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

Verbetersleutel

1. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{13}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 175 }{ \color{blue}{455} })& = & (\frac{-637}{ \color{blue}{455} }x+\frac{910}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+175& = & -637x+910 \\\Leftrightarrow & 65x \color{red}{+175} \color{blue}{-175} \color{blue}{+637x} & = & \color{red}{-637x} +910 \color{blue}{+637x} \color{blue}{-175} \\\Leftrightarrow & 702x& = & 735 \\\Leftrightarrow & \frac{702x}{ \color{red}{702} }& = & \frac{735}{702} \\\Leftrightarrow & x = \frac{245}{234} & & \\ & V = \left\{ \frac{245}{234} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -70x-84 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+70x} & = & \color{red}{-70x} -84 \color{blue}{+70x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -108 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-108}{91} \\\Leftrightarrow & x = \frac{-108}{91} & & \\ & V = \left\{ \frac{-108}{91} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{4}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+36& = & 21x+504 \\\Leftrightarrow & 28x \color{red}{+36} \color{blue}{-36} \color{blue}{-21x} & = & \color{red}{21x} +504 \color{blue}{-21x} \color{blue}{-36} \\\Leftrightarrow & 7x& = & 468 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{468}{7} \\\Leftrightarrow & x = \frac{468}{7} & & \\ & V = \left\{ \frac{468}{7} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-56}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -56x+700 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+56x} & = & \color{red}{-56x} +700 \color{blue}{+56x} \color{blue}{-80} \\\Leftrightarrow & 91x& = & 620 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{620}{91} \\\Leftrightarrow & x = \frac{620}{91} & & \\ & V = \left\{ \frac{620}{91} \right\} & \\\end{align}\)
5. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{7}{4}x-5 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 6 }{ \color{blue}{28} })& = & (\frac{49}{ \color{blue}{28} }x-\frac{140}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-6& = & 49x-140 \\\Leftrightarrow & 4x \color{red}{-6} \color{blue}{+6} \color{blue}{-49x} & = & \color{red}{49x} -140 \color{blue}{-49x} \color{blue}{+6} \\\Leftrightarrow & -45x& = & -134 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-134}{-45} \\\Leftrightarrow & x = \frac{134}{45} & & \\ & V = \left\{ \frac{134}{45} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -70x-210 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+70x} & = & \color{red}{-70x} -210 \color{blue}{+70x} \color{blue}{+12} \\\Leftrightarrow & 91x& = & -198 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-198}{91} \\\Leftrightarrow & x = \frac{-198}{91} & & \\ & V = \left\{ \frac{-198}{91} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -294x+1680 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+294x} & = & \color{red}{-294x} +1680 \color{blue}{+294x} \color{blue}{-90} \\\Leftrightarrow & 329x& = & 1590 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{1590}{329} \\\Leftrightarrow & x = \frac{1590}{329} & & \\ & V = \left\{ \frac{1590}{329} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{5}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 175x+490 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-175x} & = & \color{red}{175x} +490 \color{blue}{-175x} \color{blue}{-20} \\\Leftrightarrow & -161x& = & 470 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{470}{-161} \\\Leftrightarrow & x = \frac{-470}{161} & & \\ & V = \left\{ \frac{-470}{161} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{11}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 18 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+18& = & 33x-396 \\\Leftrightarrow & 22x \color{red}{+18} \color{blue}{-18} \color{blue}{-33x} & = & \color{red}{33x} -396 \color{blue}{-33x} \color{blue}{-18} \\\Leftrightarrow & -11x& = & -414 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-414}{-11} \\\Leftrightarrow & x = \frac{414}{11} & & \\ & V = \left\{ \frac{414}{11} \right\} & \\\end{align}\)
10. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{8}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x-\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+45& = & -168x-600 \\\Leftrightarrow & 20x \color{red}{+45} \color{blue}{-45} \color{blue}{+168x} & = & \color{red}{-168x} -600 \color{blue}{+168x} \color{blue}{-45} \\\Leftrightarrow & 188x& = & -645 \\\Leftrightarrow & \frac{188x}{ \color{red}{188} }& = & \frac{-645}{188} \\\Leftrightarrow & x = \frac{-645}{188} & & \\ & V = \left\{ \frac{-645}{188} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & -294x+420 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{+294x} & = & \color{red}{-294x} +420 \color{blue}{+294x} \color{blue}{-150} \\\Leftrightarrow & 329x& = & 270 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{270}{329} \\\Leftrightarrow & x = \frac{270}{329} & & \\ & V = \left\{ \frac{270}{329} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -24x+120 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+24x} & = & \color{red}{-24x} +120 \color{blue}{+24x} \color{blue}{-9} \\\Leftrightarrow & 29x& = & 111 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{111}{29} \\\Leftrightarrow & x = \frac{111}{29} & & \\ & V = \left\{ \frac{111}{29} \right\} & \\\end{align}\)