Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{-7}{4}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-105}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+18& = & -105x+180 \\\Leftrightarrow & 20x \color{red}{+18} \color{blue}{-18} \color{blue}{+105x} & = & \color{red}{-105x} +180 \color{blue}{+105x} \color{blue}{-18} \\\Leftrightarrow & 125x& = & 162 \\\Leftrightarrow & \frac{125x}{ \color{red}{125} }& = & \frac{162}{125} \\\Leftrightarrow & x = \frac{162}{125} & & \\ & V = \left\{ \frac{162}{125} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & 24x+96 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-24x} & = & \color{red}{24x} +96 \color{blue}{-24x} \color{blue}{+15} \\\Leftrightarrow & -8x& = & 111 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{111}{-8} \\\Leftrightarrow & x = \frac{-111}{8} & & \\ & V = \left\{ \frac{-111}{8} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & -25x-210 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{+25x} & = & \color{red}{-25x} -210 \color{blue}{+25x} \color{blue}{-8} \\\Leftrightarrow & 31x& = & -218 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{-218}{31} \\\Leftrightarrow & x = \frac{-218}{31} & & \\ & V = \left\{ \frac{-218}{31} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & -42x-90 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} -90 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 57x& = & -82 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-82}{57} \\\Leftrightarrow & x = \frac{-82}{57} & & \\ & V = \left\{ \frac{-82}{57} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x+\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -96x+1200 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+96x} & = & \color{red}{-96x} +1200 \color{blue}{+96x} \color{blue}{-45} \\\Leftrightarrow & 136x& = & 1155 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{1155}{136} \\\Leftrightarrow & x = \frac{1155}{136} & & \\ & V = \left\{ \frac{1155}{136} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 21x+210 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-21x} & = & \color{red}{21x} +210 \color{blue}{-21x} \color{blue}{+35} \\\Leftrightarrow & -15x& = & 245 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{245}{-15} \\\Leftrightarrow & x = \frac{-49}{3} & & \\ & V = \left\{ \frac{-49}{3} \right\} & \\\end{align}\)
7. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{7}{2}x-1 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{140}{ \color{blue}{40} }x-\frac{40}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+25& = & 140x-40 \\\Leftrightarrow & 8x \color{red}{+25} \color{blue}{-25} \color{blue}{-140x} & = & \color{red}{140x} -40 \color{blue}{-140x} \color{blue}{-25} \\\Leftrightarrow & -132x& = & -65 \\\Leftrightarrow & \frac{-132x}{ \color{red}{-132} }& = & \frac{-65}{-132} \\\Leftrightarrow & x = \frac{65}{132} & & \\ & V = \left\{ \frac{65}{132} \right\} & \\\end{align}\)
8. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{273}{ \color{blue}{195} }x+\frac{195}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+45& = & 273x+195 \\\Leftrightarrow & 65x \color{red}{+45} \color{blue}{-45} \color{blue}{-273x} & = & \color{red}{273x} +195 \color{blue}{-273x} \color{blue}{-45} \\\Leftrightarrow & -208x& = & 150 \\\Leftrightarrow & \frac{-208x}{ \color{red}{-208} }& = & \frac{150}{-208} \\\Leftrightarrow & x = \frac{-75}{104} & & \\ & V = \left\{ \frac{-75}{104} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 3, 12 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{3}{5}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{36}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+25& = & 36x-240 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{-36x} & = & \color{red}{36x} -240 \color{blue}{-36x} \color{blue}{-25} \\\Leftrightarrow & -16x& = & -265 \\\Leftrightarrow & \frac{-16x}{ \color{red}{-16} }& = & \frac{-265}{-16} \\\Leftrightarrow & x = \frac{265}{16} & & \\ & V = \left\{ \frac{265}{16} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{70}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 70x+42 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-70x} & = & \color{red}{70x} +42 \color{blue}{-70x} \color{blue}{+12} \\\Leftrightarrow & -49x& = & 54 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{54}{-49} \\\Leftrightarrow & x = \frac{-54}{49} & & \\ & V = \left\{ \frac{-54}{49} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 6x+90 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} +90 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & 3x& = & 86 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{86}{3} \\\Leftrightarrow & x = \frac{86}{3} & & \\ & V = \left\{ \frac{86}{3} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & -140x-336 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{+140x} & = & \color{red}{-140x} -336 \color{blue}{+140x} \color{blue}{-36} \\\Leftrightarrow & 161x& = & -372 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-372}{161} \\\Leftrightarrow & x = \frac{-372}{161} & & \\ & V = \left\{ \frac{-372}{161} \right\} & \\\end{align}\)