Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{5}{9}=\frac{6}{5}x+4\)
2. \(\frac{x}{7}-\frac{2}{9}=\frac{1}{2}x-4\)
3. \(\frac{x}{5}-\frac{2}{13}=\frac{-2}{3}x-4\)
4. \(\frac{x}{3}-\frac{5}{13}=\frac{1}{2}x+2\)
5. \(\frac{x}{5}-\frac{2}{11}=\frac{5}{6}x-8\)
6. \(\frac{x}{4}-\frac{4}{13}=\frac{1}{5}x+3\)
7. \(\frac{x}{7}+\frac{4}{7}=\frac{1}{6}x-7\)
8. \(\frac{x}{7}-\frac{2}{11}=\frac{5}{6}x-4\)
9. \(\frac{x}{2}+\frac{3}{16}=\frac{2}{3}x-5\)
10. \(\frac{x}{2}+\frac{3}{11}=\frac{4}{3}x+6\)
11. \(\frac{x}{7}-\frac{3}{13}=\frac{-7}{4}x-7\)
12. \(\frac{x}{6}+\frac{4}{7}=\frac{1}{5}x+7\)

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+50& = & 108x+360 \\\Leftrightarrow & 45x \color{red}{+50} \color{blue}{-50} \color{blue}{-108x} & = & \color{red}{108x} +360 \color{blue}{-108x} \color{blue}{-50} \\\Leftrightarrow & -63x& = & 310 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{310}{-63} \\\Leftrightarrow & x = \frac{-310}{63} & & \\ & V = \left\{ \frac{-310}{63} \right\} & \\\end{align}\)
2. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 28 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x-\frac{504}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-28& = & 63x-504 \\\Leftrightarrow & 18x \color{red}{-28} \color{blue}{+28} \color{blue}{-63x} & = & \color{red}{63x} -504 \color{blue}{-63x} \color{blue}{+28} \\\Leftrightarrow & -45x& = & -476 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-476}{-45} \\\Leftrightarrow & x = \frac{476}{45} & & \\ & V = \left\{ \frac{476}{45} \right\} & \\\end{align}\)
3. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 30 }{ \color{blue}{195} })& = & (\frac{-130}{ \color{blue}{195} }x-\frac{780}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-30& = & -130x-780 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{+130x} & = & \color{red}{-130x} -780 \color{blue}{+130x} \color{blue}{+30} \\\Leftrightarrow & 169x& = & -750 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-750}{169} \\\Leftrightarrow & x = \frac{-750}{169} & & \\ & V = \left\{ \frac{-750}{169} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-30& = & 39x+156 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-39x} & = & \color{red}{39x} +156 \color{blue}{-39x} \color{blue}{+30} \\\Leftrightarrow & -13x& = & 186 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{186}{-13} \\\Leftrightarrow & x = \frac{-186}{13} & & \\ & V = \left\{ \frac{-186}{13} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{5}{6}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{275}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-60& = & 275x-2640 \\\Leftrightarrow & 66x \color{red}{-60} \color{blue}{+60} \color{blue}{-275x} & = & \color{red}{275x} -2640 \color{blue}{-275x} \color{blue}{+60} \\\Leftrightarrow & -209x& = & -2580 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-2580}{-209} \\\Leftrightarrow & x = \frac{2580}{209} & & \\ & V = \left\{ \frac{2580}{209} \right\} & \\\end{align}\)
6. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{52}{ \color{blue}{260} }x+\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-80& = & 52x+780 \\\Leftrightarrow & 65x \color{red}{-80} \color{blue}{+80} \color{blue}{-52x} & = & \color{red}{52x} +780 \color{blue}{-52x} \color{blue}{+80} \\\Leftrightarrow & 13x& = & 860 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{860}{13} \\\Leftrightarrow & x = \frac{860}{13} & & \\ & V = \left\{ \frac{860}{13} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+24& = & 7x-294 \\\Leftrightarrow & 6x \color{red}{+24} \color{blue}{-24} \color{blue}{-7x} & = & \color{red}{7x} -294 \color{blue}{-7x} \color{blue}{-24} \\\Leftrightarrow & -x& = & -318 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-318}{-1} \\\Leftrightarrow & x = 318 & & \\ & V = \left\{ 318 \right\} & \\\end{align}\)
8. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{5}{6}x-4 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 84 }{ \color{blue}{462} })& = & (\frac{385}{ \color{blue}{462} }x-\frac{1848}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-84& = & 385x-1848 \\\Leftrightarrow & 66x \color{red}{-84} \color{blue}{+84} \color{blue}{-385x} & = & \color{red}{385x} -1848 \color{blue}{-385x} \color{blue}{+84} \\\Leftrightarrow & -319x& = & -1764 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{-1764}{-319} \\\Leftrightarrow & x = \frac{1764}{319} & & \\ & V = \left\{ \frac{1764}{319} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{2}{3}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{32}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 32x-240 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-32x} & = & \color{red}{32x} -240 \color{blue}{-32x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & -249 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-249}{-8} \\\Leftrightarrow & x = \frac{249}{8} & & \\ & V = \left\{ \frac{249}{8} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{11}& = & \frac{4}{3}x+6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 18 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x+\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+18& = & 88x+396 \\\Leftrightarrow & 33x \color{red}{+18} \color{blue}{-18} \color{blue}{-88x} & = & \color{red}{88x} +396 \color{blue}{-88x} \color{blue}{-18} \\\Leftrightarrow & -55x& = & 378 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{378}{-55} \\\Leftrightarrow & x = \frac{-378}{55} & & \\ & V = \left\{ \frac{-378}{55} \right\} & \\\end{align}\)
11. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-7}{4}x-7 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 84 }{ \color{blue}{364} })& = & (\frac{-637}{ \color{blue}{364} }x-\frac{2548}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-84& = & -637x-2548 \\\Leftrightarrow & 52x \color{red}{-84} \color{blue}{+84} \color{blue}{+637x} & = & \color{red}{-637x} -2548 \color{blue}{+637x} \color{blue}{+84} \\\Leftrightarrow & 689x& = & -2464 \\\Leftrightarrow & \frac{689x}{ \color{red}{689} }& = & \frac{-2464}{689} \\\Leftrightarrow & x = \frac{-2464}{689} & & \\ & V = \left\{ \frac{-2464}{689} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 42x+1470 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-42x} & = & \color{red}{42x} +1470 \color{blue}{-42x} \color{blue}{-120} \\\Leftrightarrow & -7x& = & 1350 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{1350}{-7} \\\Leftrightarrow & x = \frac{-1350}{7} & & \\ & V = \left\{ \frac{-1350}{7} \right\} & \\\end{align}\)