Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}+\frac{4}{15}=\frac{5}{4}x+4\)
2. \(\frac{x}{3}-\frac{3}{7}=\frac{3}{2}x-4\)
3. \(\frac{x}{6}-\frac{5}{6}=\frac{-2}{5}x-8\)
4. \(\frac{x}{5}+\frac{3}{7}=\frac{7}{6}x-5\)
5. \(\frac{x}{7}+\frac{3}{7}=\frac{-5}{3}x+4\)
6. \(\frac{x}{5}-\frac{3}{13}=\frac{-3}{4}x+3\)
7. \(\frac{x}{6}+\frac{3}{10}=\frac{6}{5}x-5\)
8. \(\frac{x}{4}+\frac{2}{11}=\frac{-2}{3}x+2\)
9. \(\frac{x}{2}-\frac{5}{16}=\frac{-4}{5}x+5\)
10. \(\frac{x}{3}-\frac{2}{11}=\frac{1}{4}x+3\)
11. \(\frac{x}{6}+\frac{3}{16}=\frac{6}{5}x+1\)
12. \(\frac{x}{2}-\frac{5}{8}=\frac{2}{3}x-7\)

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{5}{4}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+16& = & 75x+240 \\\Leftrightarrow & 20x \color{red}{+16} \color{blue}{-16} \color{blue}{-75x} & = & \color{red}{75x} +240 \color{blue}{-75x} \color{blue}{-16} \\\Leftrightarrow & -55x& = & 224 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{224}{-55} \\\Leftrightarrow & x = \frac{-224}{55} & & \\ & V = \left\{ \frac{-224}{55} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{3}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 63x-168 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-63x} & = & \color{red}{63x} -168 \color{blue}{-63x} \color{blue}{+18} \\\Leftrightarrow & -49x& = & -150 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-150}{-49} \\\Leftrightarrow & x = \frac{150}{49} & & \\ & V = \left\{ \frac{150}{49} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & -12x-240 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+12x} & = & \color{red}{-12x} -240 \color{blue}{+12x} \color{blue}{+25} \\\Leftrightarrow & 17x& = & -215 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-215}{17} \\\Leftrightarrow & x = \frac{-215}{17} & & \\ & V = \left\{ \frac{-215}{17} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{7}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & 245x-1050 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{-245x} & = & \color{red}{245x} -1050 \color{blue}{-245x} \color{blue}{-90} \\\Leftrightarrow & -203x& = & -1140 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{-1140}{-203} \\\Leftrightarrow & x = \frac{1140}{203} & & \\ & V = \left\{ \frac{1140}{203} \right\} & \\\end{align}\)
5. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 9 }{ \color{blue}{21} })& = & (\frac{-35}{ \color{blue}{21} }x+\frac{84}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+9& = & -35x+84 \\\Leftrightarrow & 3x \color{red}{+9} \color{blue}{-9} \color{blue}{+35x} & = & \color{red}{-35x} +84 \color{blue}{+35x} \color{blue}{-9} \\\Leftrightarrow & 38x& = & 75 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{75}{38} \\\Leftrightarrow & x = \frac{75}{38} & & \\ & V = \left\{ \frac{75}{38} \right\} & \\\end{align}\)
6. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{-195}{ \color{blue}{260} }x+\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-60& = & -195x+780 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{+195x} & = & \color{red}{-195x} +780 \color{blue}{+195x} \color{blue}{+60} \\\Leftrightarrow & 247x& = & 840 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{840}{247} \\\Leftrightarrow & x = \frac{840}{247} & & \\ & V = \left\{ \frac{840}{247} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 36x-150 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-36x} & = & \color{red}{36x} -150 \color{blue}{-36x} \color{blue}{-9} \\\Leftrightarrow & -31x& = & -159 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-159}{-31} \\\Leftrightarrow & x = \frac{159}{31} & & \\ & V = \left\{ \frac{159}{31} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x+\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -88x+264 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+88x} & = & \color{red}{-88x} +264 \color{blue}{+88x} \color{blue}{-24} \\\Leftrightarrow & 121x& = & 240 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{240}{121} \\\Leftrightarrow & x = \frac{240}{121} & & \\ & V = \left\{ \frac{240}{121} \right\} & \\\end{align}\)
9. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & -64x+400 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{+64x} & = & \color{red}{-64x} +400 \color{blue}{+64x} \color{blue}{+25} \\\Leftrightarrow & 104x& = & 425 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{425}{104} \\\Leftrightarrow & x = \frac{425}{104} & & \\ & V = \left\{ \frac{425}{104} \right\} & \\\end{align}\)
10. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x+\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-24& = & 33x+396 \\\Leftrightarrow & 44x \color{red}{-24} \color{blue}{+24} \color{blue}{-33x} & = & \color{red}{33x} +396 \color{blue}{-33x} \color{blue}{+24} \\\Leftrightarrow & 11x& = & 420 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{420}{11} \\\Leftrightarrow & x = \frac{420}{11} & & \\ & V = \left\{ \frac{420}{11} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & 288x+240 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-288x} & = & \color{red}{288x} +240 \color{blue}{-288x} \color{blue}{-45} \\\Leftrightarrow & -248x& = & 195 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{195}{-248} \\\Leftrightarrow & x = \frac{-195}{248} & & \\ & V = \left\{ \frac{-195}{248} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{2}{3}x-7 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{16}{ \color{blue}{24} }x-\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & 16x-168 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} -168 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & -4x& = & -153 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-153}{-4} \\\Leftrightarrow & x = \frac{153}{4} & & \\ & V = \left\{ \frac{153}{4} \right\} & \\\end{align}\)