Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 4, 12 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-25& = & 72x+180 \\\Leftrightarrow & 15x \color{red}{-25} \color{blue}{+25} \color{blue}{-72x} & = & \color{red}{72x} +180 \color{blue}{-72x} \color{blue}{+25} \\\Leftrightarrow & -57x& = & 205 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{205}{-57} \\\Leftrightarrow & x = \frac{-205}{57} & & \\ & V = \left\{ \frac{-205}{57} \right\} & \\\end{align}\)
2. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{6}{5}x-6 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{216}{ \color{blue}{180} }x-\frac{1080}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+40& = & 216x-1080 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{-216x} & = & \color{red}{216x} -1080 \color{blue}{-216x} \color{blue}{-40} \\\Leftrightarrow & -171x& = & -1120 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{-1120}{-171} \\\Leftrightarrow & x = \frac{1120}{171} & & \\ & V = \left\{ \frac{1120}{171} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{3}{5}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{36}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & 36x+60 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-36x} & = & \color{red}{36x} +60 \color{blue}{-36x} \color{blue}{-25} \\\Leftrightarrow & -26x& = & 35 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{35}{-26} \\\Leftrightarrow & x = \frac{-35}{26} & & \\ & V = \left\{ \frac{-35}{26} \right\} & \\\end{align}\)
4. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{21}{ \color{blue}{105} }x-\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-45& = & 21x-210 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{+45} \\\Leftrightarrow & 14x& = & -165 \\\Leftrightarrow & \frac{14x}{ \color{red}{14} }& = & \frac{-165}{14} \\\Leftrightarrow & x = \frac{-165}{14} & & \\ & V = \left\{ \frac{-165}{14} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 6 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+25& = & 36x+90 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{-36x} & = & \color{red}{36x} +90 \color{blue}{-36x} \color{blue}{-25} \\\Leftrightarrow & -21x& = & 65 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{65}{-21} \\\Leftrightarrow & x = \frac{-65}{21} & & \\ & V = \left\{ \frac{-65}{21} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{30}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & 30x+120 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{-30x} & = & \color{red}{30x} +120 \color{blue}{-30x} \color{blue}{-25} \\\Leftrightarrow & -18x& = & 95 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{95}{-18} \\\Leftrightarrow & x = \frac{-95}{18} & & \\ & V = \left\{ \frac{-95}{18} \right\} & \\\end{align}\)
7. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{7}{4}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{140}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 140x-400 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-140x} & = & \color{red}{140x} -400 \color{blue}{-140x} \color{blue}{+25} \\\Leftrightarrow & -124x& = & -375 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{-375}{-124} \\\Leftrightarrow & x = \frac{375}{124} & & \\ & V = \left\{ \frac{375}{124} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & -56x-560 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+56x} & = & \color{red}{-56x} -560 \color{blue}{+56x} \color{blue}{-40} \\\Leftrightarrow & 91x& = & -600 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-600}{91} \\\Leftrightarrow & x = \frac{-600}{91} & & \\ & V = \left\{ \frac{-600}{91} \right\} & \\\end{align}\)
9. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & 6x+54 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{-6x} & = & \color{red}{6x} +54 \color{blue}{-6x} \color{blue}{-10} \\\Leftrightarrow & 3x& = & 44 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{44}{3} \\\Leftrightarrow & x = \frac{44}{3} & & \\ & V = \left\{ \frac{44}{3} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{7}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 42x+120 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-42x} & = & \color{red}{42x} +120 \color{blue}{-42x} \color{blue}{-9} \\\Leftrightarrow & -37x& = & 111 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{111}{-37} \\\Leftrightarrow & x = -3 & & \\ & V = \left\{ -3 \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x-\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-120& = & -275x-2310 \\\Leftrightarrow & 66x \color{red}{-120} \color{blue}{+120} \color{blue}{+275x} & = & \color{red}{-275x} -2310 \color{blue}{+275x} \color{blue}{+120} \\\Leftrightarrow & 341x& = & -2190 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{-2190}{341} \\\Leftrightarrow & x = \frac{-2190}{341} & & \\ & V = \left\{ \frac{-2190}{341} \right\} & \\\end{align}\)
12. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{-14}{ \color{blue}{10} }x+\frac{30}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x+3& = & -14x+30 \\\Leftrightarrow & 5x \color{red}{+3} \color{blue}{-3} \color{blue}{+14x} & = & \color{red}{-14x} +30 \color{blue}{+14x} \color{blue}{-3} \\\Leftrightarrow & 19x& = & 27 \\\Leftrightarrow & \frac{19x}{ \color{red}{19} }& = & \frac{27}{19} \\\Leftrightarrow & x = \frac{27}{19} & & \\ & V = \left\{ \frac{27}{19} \right\} & \\\end{align}\)