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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & 36x-120 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{-36x} & = & \color{red}{36x} -120 \color{blue}{-36x} \color{blue}{-25} \\\Leftrightarrow & -31x& = & -145 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-145}{-31} \\\Leftrightarrow & x = \frac{145}{31} & & \\ & V = \left\{ \frac{145}{31} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 21x-42 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-21x} & = & \color{red}{21x} -42 \color{blue}{-21x} \color{blue}{-18} \\\Leftrightarrow & -7x& = & -60 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-60}{-7} \\\Leftrightarrow & x = \frac{60}{7} & & \\ & V = \left\{ \frac{60}{7} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & 196x-504 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{-196x} & = & \color{red}{196x} -504 \color{blue}{-196x} \color{blue}{-60} \\\Leftrightarrow & -175x& = & -564 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-564}{-175} \\\Leftrightarrow & x = \frac{564}{175} & & \\ & V = \left\{ \frac{564}{175} \right\} & \\\end{align}\)
4. \(\text{12 is het kleinste gemene veelvoud van 3, 6 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{3}{4}x-2 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{9}{ \color{blue}{12} }x-\frac{24}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+10& = & 9x-24 \\\Leftrightarrow & 4x \color{red}{+10} \color{blue}{-10} \color{blue}{-9x} & = & \color{red}{9x} -24 \color{blue}{-9x} \color{blue}{-10} \\\Leftrightarrow & -5x& = & -34 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-34}{-5} \\\Leftrightarrow & x = \frac{34}{5} & & \\ & V = \left\{ \frac{34}{5} \right\} & \\\end{align}\)
5. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 45 }{ \color{blue}{165} })& = & (\frac{385}{ \color{blue}{165} }x-\frac{990}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+45& = & 385x-990 \\\Leftrightarrow & 33x \color{red}{+45} \color{blue}{-45} \color{blue}{-385x} & = & \color{red}{385x} -990 \color{blue}{-385x} \color{blue}{-45} \\\Leftrightarrow & -352x& = & -1035 \\\Leftrightarrow & \frac{-352x}{ \color{red}{-352} }& = & \frac{-1035}{-352} \\\Leftrightarrow & x = \frac{1035}{352} & & \\ & V = \left\{ \frac{1035}{352} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{11}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-90& = & 396x-2640 \\\Leftrightarrow & 55x \color{red}{-90} \color{blue}{+90} \color{blue}{-396x} & = & \color{red}{396x} -2640 \color{blue}{-396x} \color{blue}{+90} \\\Leftrightarrow & -341x& = & -2550 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-2550}{-341} \\\Leftrightarrow & x = \frac{2550}{341} & & \\ & V = \left\{ \frac{2550}{341} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 5, 6 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{5}{4}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 50 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+50& = & 75x-240 \\\Leftrightarrow & 12x \color{red}{+50} \color{blue}{-50} \color{blue}{-75x} & = & \color{red}{75x} -240 \color{blue}{-75x} \color{blue}{-50} \\\Leftrightarrow & -63x& = & -290 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{-290}{-63} \\\Leftrightarrow & x = \frac{290}{63} & & \\ & V = \left\{ \frac{290}{63} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{84}{ \color{blue}{24} }x-\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 84x-168 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-84x} & = & \color{red}{84x} -168 \color{blue}{-84x} \color{blue}{+9} \\\Leftrightarrow & -76x& = & -159 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{-159}{-76} \\\Leftrightarrow & x = \frac{159}{76} & & \\ & V = \left\{ \frac{159}{76} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{5}{6}x+8 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{100}{ \color{blue}{120} }x+\frac{960}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x+75& = & 100x+960 \\\Leftrightarrow & 24x \color{red}{+75} \color{blue}{-75} \color{blue}{-100x} & = & \color{red}{100x} +960 \color{blue}{-100x} \color{blue}{-75} \\\Leftrightarrow & -76x& = & 885 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{885}{-76} \\\Leftrightarrow & x = \frac{-885}{76} & & \\ & V = \left\{ \frac{-885}{76} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{13}& = & \frac{7}{6}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{455}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+150& = & 455x-1170 \\\Leftrightarrow & 78x \color{red}{+150} \color{blue}{-150} \color{blue}{-455x} & = & \color{red}{455x} -1170 \color{blue}{-455x} \color{blue}{-150} \\\Leftrightarrow & -377x& = & -1320 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{-1320}{-377} \\\Leftrightarrow & x = \frac{1320}{377} & & \\ & V = \left\{ \frac{1320}{377} \right\} & \\\end{align}\)
11. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{12}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-8}{ \color{blue}{12} }x+\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x-5& = & -8x+72 \\\Leftrightarrow & 6x \color{red}{-5} \color{blue}{+5} \color{blue}{+8x} & = & \color{red}{-8x} +72 \color{blue}{+8x} \color{blue}{+5} \\\Leftrightarrow & 14x& = & 77 \\\Leftrightarrow & \frac{14x}{ \color{red}{14} }& = & \frac{77}{14} \\\Leftrightarrow & x = \frac{11}{2} & & \\ & V = \left\{ \frac{11}{2} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{5}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 140x+672 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-140x} & = & \color{red}{140x} +672 \color{blue}{-140x} \color{blue}{-24} \\\Leftrightarrow & -119x& = & 648 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{648}{-119} \\\Leftrightarrow & x = \frac{-648}{119} & & \\ & V = \left\{ \frac{-648}{119} \right\} & \\\end{align}\)