Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}-\frac{4}{15}=\frac{7}{3}x+6\)
2. \(\frac{x}{3}-\frac{4}{11}=\frac{6}{5}x+8\)
3. \(\frac{x}{4}+\frac{2}{7}=\frac{2}{5}x-4\)
4. \(\frac{x}{4}-\frac{3}{10}=\frac{-5}{3}x+7\)
5. \(\frac{x}{3}+\frac{4}{7}=\frac{7}{5}x+7\)
6. \(\frac{x}{5}-\frac{2}{7}=\frac{1}{3}x+1\)
7. \(\frac{x}{6}+\frac{5}{12}=\frac{-7}{5}x+3\)
8. \(\frac{x}{5}+\frac{3}{16}=\frac{5}{2}x-8\)
9. \(\frac{x}{7}+\frac{2}{9}=\frac{1}{6}x+7\)
10. \(\frac{x}{3}-\frac{5}{14}=\frac{-4}{5}x-8\)
11. \(\frac{x}{6}-\frac{3}{10}=\frac{3}{5}x-7\)
12. \(\frac{x}{7}+\frac{2}{15}=\frac{1}{2}x-4\)

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

Verbetersleutel

1. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{7}{3}x+6 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 28 }{ \color{blue}{105} })& = & (\frac{245}{ \color{blue}{105} }x+\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-28& = & 245x+630 \\\Leftrightarrow & 15x \color{red}{-28} \color{blue}{+28} \color{blue}{-245x} & = & \color{red}{245x} +630 \color{blue}{-245x} \color{blue}{+28} \\\Leftrightarrow & -230x& = & 658 \\\Leftrightarrow & \frac{-230x}{ \color{red}{-230} }& = & \frac{658}{-230} \\\Leftrightarrow & x = \frac{-329}{115} & & \\ & V = \left\{ \frac{-329}{115} \right\} & \\\end{align}\)
2. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 60 }{ \color{blue}{165} })& = & (\frac{198}{ \color{blue}{165} }x+\frac{1320}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-60& = & 198x+1320 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-198x} & = & \color{red}{198x} +1320 \color{blue}{-198x} \color{blue}{+60} \\\Leftrightarrow & -143x& = & 1380 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{1380}{-143} \\\Leftrightarrow & x = \frac{-1380}{143} & & \\ & V = \left\{ \frac{-1380}{143} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{2}{5}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{56}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 56x-560 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-56x} & = & \color{red}{56x} -560 \color{blue}{-56x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & -600 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-600}{-21} \\\Leftrightarrow & x = \frac{200}{7} & & \\ & V = \left\{ \frac{200}{7} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & -100x+420 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{+100x} & = & \color{red}{-100x} +420 \color{blue}{+100x} \color{blue}{+18} \\\Leftrightarrow & 115x& = & 438 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{438}{115} \\\Leftrightarrow & x = \frac{438}{115} & & \\ & V = \left\{ \frac{438}{115} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{147}{ \color{blue}{105} }x+\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+60& = & 147x+735 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-147x} & = & \color{red}{147x} +735 \color{blue}{-147x} \color{blue}{-60} \\\Leftrightarrow & -112x& = & 675 \\\Leftrightarrow & \frac{-112x}{ \color{red}{-112} }& = & \frac{675}{-112} \\\Leftrightarrow & x = \frac{-675}{112} & & \\ & V = \left\{ \frac{-675}{112} \right\} & \\\end{align}\)
6. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 30 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x+\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-30& = & 35x+105 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-35x} & = & \color{red}{35x} +105 \color{blue}{-35x} \color{blue}{+30} \\\Leftrightarrow & -14x& = & 135 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{135}{-14} \\\Leftrightarrow & x = \frac{-135}{14} & & \\ & V = \left\{ \frac{-135}{14} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & -84x+180 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{+84x} & = & \color{red}{-84x} +180 \color{blue}{+84x} \color{blue}{-25} \\\Leftrightarrow & 94x& = & 155 \\\Leftrightarrow & \frac{94x}{ \color{red}{94} }& = & \frac{155}{94} \\\Leftrightarrow & x = \frac{155}{94} & & \\ & V = \left\{ \frac{155}{94} \right\} & \\\end{align}\)
8. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{200}{ \color{blue}{80} }x-\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 200x-640 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-200x} & = & \color{red}{200x} -640 \color{blue}{-200x} \color{blue}{-15} \\\Leftrightarrow & -184x& = & -655 \\\Leftrightarrow & \frac{-184x}{ \color{red}{-184} }& = & \frac{-655}{-184} \\\Leftrightarrow & x = \frac{655}{184} & & \\ & V = \left\{ \frac{655}{184} \right\} & \\\end{align}\)
9. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{1}{6}x+7 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{21}{ \color{blue}{126} }x+\frac{882}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & 21x+882 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{-21x} & = & \color{red}{21x} +882 \color{blue}{-21x} \color{blue}{-28} \\\Leftrightarrow & -3x& = & 854 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{854}{-3} \\\Leftrightarrow & x = \frac{-854}{3} & & \\ & V = \left\{ \frac{-854}{3} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 3, 14 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{14}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{70x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 70x-75& = & -168x-1680 \\\Leftrightarrow & 70x \color{red}{-75} \color{blue}{+75} \color{blue}{+168x} & = & \color{red}{-168x} -1680 \color{blue}{+168x} \color{blue}{+75} \\\Leftrightarrow & 238x& = & -1605 \\\Leftrightarrow & \frac{238x}{ \color{red}{238} }& = & \frac{-1605}{238} \\\Leftrightarrow & x = \frac{-1605}{238} & & \\ & V = \left\{ \frac{-1605}{238} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{3}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & 18x-210 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{-18x} & = & \color{red}{18x} -210 \color{blue}{-18x} \color{blue}{+9} \\\Leftrightarrow & -13x& = & -201 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-201}{-13} \\\Leftrightarrow & x = \frac{201}{13} & & \\ & V = \left\{ \frac{201}{13} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{105}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & 105x-840 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{-105x} & = & \color{red}{105x} -840 \color{blue}{-105x} \color{blue}{-28} \\\Leftrightarrow & -75x& = & -868 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-868}{-75} \\\Leftrightarrow & x = \frac{868}{75} & & \\ & V = \left\{ \frac{868}{75} \right\} & \\\end{align}\)