Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{5}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{70}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 70x+84 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-70x} & = & \color{red}{70x} +84 \color{blue}{-70x} \color{blue}{-12} \\\Leftrightarrow & -49x& = & 72 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{72}{-49} \\\Leftrightarrow & x = \frac{-72}{49} & & \\ & V = \left\{ \frac{-72}{49} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 15x+180 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-15x} & = & \color{red}{15x} +180 \color{blue}{-15x} \color{blue}{+9} \\\Leftrightarrow & -5x& = & 189 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{189}{-5} \\\Leftrightarrow & x = \frac{-189}{5} & & \\ & V = \left\{ \frac{-189}{5} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-60& = & -84x+210 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+84x} & = & \color{red}{-84x} +210 \color{blue}{+84x} \color{blue}{+60} \\\Leftrightarrow & 119x& = & 270 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{270}{119} \\\Leftrightarrow & x = \frac{270}{119} & & \\ & V = \left\{ \frac{270}{119} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 36x+60 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-36x} & = & \color{red}{36x} +60 \color{blue}{-36x} \color{blue}{+8} \\\Leftrightarrow & -31x& = & 68 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{68}{-31} \\\Leftrightarrow & x = \frac{-68}{31} & & \\ & V = \left\{ \frac{-68}{31} \right\} & \\\end{align}\)
5. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{195}{ \color{blue}{78} }x-\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 195x-624 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-195x} & = & \color{red}{195x} -624 \color{blue}{-195x} \color{blue}{-12} \\\Leftrightarrow & -169x& = & -636 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-636}{-169} \\\Leftrightarrow & x = \frac{636}{169} & & \\ & V = \left\{ \frac{636}{169} \right\} & \\\end{align}\)
6. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{1}{4}x+7 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x+\frac{1820}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-80& = & 65x+1820 \\\Leftrightarrow & 52x \color{red}{-80} \color{blue}{+80} \color{blue}{-65x} & = & \color{red}{65x} +1820 \color{blue}{-65x} \color{blue}{+80} \\\Leftrightarrow & -13x& = & 1900 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{1900}{-13} \\\Leftrightarrow & x = \frac{-1900}{13} & & \\ & V = \left\{ \frac{-1900}{13} \right\} & \\\end{align}\)
7. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 4 }{ \color{blue}{15} })& = & (\frac{3}{ \color{blue}{15} }x+\frac{30}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+4& = & 3x+30 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-3x} & = & \color{red}{3x} +30 \color{blue}{-3x} \color{blue}{-4} \\\Leftrightarrow & 2x& = & 26 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{26}{2} \\\Leftrightarrow & x = 13 & & \\ & V = \left\{ 13 \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & -140x-84 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{+140x} & = & \color{red}{-140x} -84 \color{blue}{+140x} \color{blue}{-36} \\\Leftrightarrow & 161x& = & -120 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-120}{161} \\\Leftrightarrow & x = \frac{-120}{161} & & \\ & V = \left\{ \frac{-120}{161} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{-3}{4}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & -45x-480 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{+45x} & = & \color{red}{-45x} -480 \color{blue}{+45x} \color{blue}{+25} \\\Leftrightarrow & 57x& = & -455 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-455}{57} \\\Leftrightarrow & x = \frac{-455}{57} & & \\ & V = \left\{ \frac{-455}{57} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 56x+84 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-56x} & = & \color{red}{56x} +84 \color{blue}{-56x} \color{blue}{+18} \\\Leftrightarrow & -35x& = & 102 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{102}{-35} \\\Leftrightarrow & x = \frac{-102}{35} & & \\ & V = \left\{ \frac{-102}{35} \right\} & \\\end{align}\)
11. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{4}{5}x-2 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{208}{ \color{blue}{260} }x-\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+40& = & 208x-520 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{-208x} & = & \color{red}{208x} -520 \color{blue}{-208x} \color{blue}{-40} \\\Leftrightarrow & -143x& = & -560 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-560}{-143} \\\Leftrightarrow & x = \frac{560}{143} & & \\ & V = \left\{ \frac{560}{143} \right\} & \\\end{align}\)
12. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x+\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & -96x+720 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{+96x} & = & \color{red}{-96x} +720 \color{blue}{+96x} \color{blue}{+75} \\\Leftrightarrow & 136x& = & 795 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{795}{136} \\\Leftrightarrow & x = \frac{795}{136} & & \\ & V = \left\{ \frac{795}{136} \right\} & \\\end{align}\)