Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{2}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 84x-1260 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-84x} & = & \color{red}{84x} -1260 \color{blue}{-84x} \color{blue}{+90} \\\Leftrightarrow & -49x& = & -1170 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-1170}{-49} \\\Leftrightarrow & x = \frac{1170}{49} & & \\ & V = \left\{ \frac{1170}{49} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-4}{5}x-4 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x-\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -192x-960 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+192x} & = & \color{red}{-192x} -960 \color{blue}{+192x} \color{blue}{-45} \\\Leftrightarrow & 232x& = & -1005 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{-1005}{232} \\\Leftrightarrow & x = \frac{-1005}{232} & & \\ & V = \left\{ \frac{-1005}{232} \right\} & \\\end{align}\)
3. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 12 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x+\frac{112}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+12& = & 7x+112 \\\Leftrightarrow & 4x \color{red}{+12} \color{blue}{-12} \color{blue}{-7x} & = & \color{red}{7x} +112 \color{blue}{-7x} \color{blue}{-12} \\\Leftrightarrow & -3x& = & 100 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{100}{-3} \\\Leftrightarrow & x = \frac{-100}{3} & & \\ & V = \left\{ \frac{-100}{3} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 5x-240 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-5x} & = & \color{red}{5x} -240 \color{blue}{-5x} \color{blue}{-25} \\\Leftrightarrow & x& = & -265 \\ & V = \left\{ -265 \right\} & \\\end{align}\)
5. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{3}{2}x+2 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 28 }{ \color{blue}{154} })& = & (\frac{231}{ \color{blue}{154} }x+\frac{308}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-28& = & 231x+308 \\\Leftrightarrow & 22x \color{red}{-28} \color{blue}{+28} \color{blue}{-231x} & = & \color{red}{231x} +308 \color{blue}{-231x} \color{blue}{+28} \\\Leftrightarrow & -209x& = & 336 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{336}{-209} \\\Leftrightarrow & x = \frac{-336}{209} & & \\ & V = \left\{ \frac{-336}{209} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -12x+120 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+12x} & = & \color{red}{-12x} +120 \color{blue}{+12x} \color{blue}{-9} \\\Leftrightarrow & 17x& = & 111 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{111}{17} \\\Leftrightarrow & x = \frac{111}{17} & & \\ & V = \left\{ \frac{111}{17} \right\} & \\\end{align}\)
7. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{32}{ \color{blue}{24} }x+\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+15& = & 32x+192 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{-32x} & = & \color{red}{32x} +192 \color{blue}{-32x} \color{blue}{-15} \\\Leftrightarrow & -20x& = & 177 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{177}{-20} \\\Leftrightarrow & x = \frac{-177}{20} & & \\ & V = \left\{ \frac{-177}{20} \right\} & \\\end{align}\)
8. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{8}{ \color{blue}{6} }x-\frac{18}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 8x-18 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-8x} & = & \color{red}{8x} -18 \color{blue}{-8x} \color{blue}{-5} \\\Leftrightarrow & -5x& = & -23 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-23}{-5} \\\Leftrightarrow & x = \frac{23}{5} & & \\ & V = \left\{ \frac{23}{5} \right\} & \\\end{align}\)
9. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{-3}{4}x-8 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }- \frac{ 112 }{ \color{blue}{252} })& = & (\frac{-189}{ \color{blue}{252} }x-\frac{2016}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x-112& = & -189x-2016 \\\Leftrightarrow & 36x \color{red}{-112} \color{blue}{+112} \color{blue}{+189x} & = & \color{red}{-189x} -2016 \color{blue}{+189x} \color{blue}{+112} \\\Leftrightarrow & 225x& = & -1904 \\\Leftrightarrow & \frac{225x}{ \color{red}{225} }& = & \frac{-1904}{225} \\\Leftrightarrow & x = \frac{-1904}{225} & & \\ & V = \left\{ \frac{-1904}{225} \right\} & \\\end{align}\)
10. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{12}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{6}{ \color{blue}{12} }x+\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-5& = & 6x+96 \\\Leftrightarrow & 4x \color{red}{-5} \color{blue}{+5} \color{blue}{-6x} & = & \color{red}{6x} +96 \color{blue}{-6x} \color{blue}{+5} \\\Leftrightarrow & -2x& = & 101 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{101}{-2} \\\Leftrightarrow & x = \frac{-101}{2} & & \\ & V = \left\{ \frac{-101}{2} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-210 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -186 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-186}{-7} \\\Leftrightarrow & x = \frac{186}{7} & & \\ & V = \left\{ \frac{186}{7} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{7}{5}x+4 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{462}{ \color{blue}{330} }x+\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & 462x+1320 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-462x} & = & \color{red}{462x} +1320 \color{blue}{-462x} \color{blue}{+60} \\\Leftrightarrow & -407x& = & 1380 \\\Leftrightarrow & \frac{-407x}{ \color{red}{-407} }& = & \frac{1380}{-407} \\\Leftrightarrow & x = \frac{-1380}{407} & & \\ & V = \left\{ \frac{-1380}{407} \right\} & \\\end{align}\)