Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{9}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+50& = & -126x+180 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{+126x} & = & \color{red}{-126x} +180 \color{blue}{+126x} \color{blue}{-50} \\\Leftrightarrow & 141x& = & 130 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{130}{141} \\\Leftrightarrow & x = \frac{130}{141} & & \\ & V = \left\{ \frac{130}{141} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{7}{6}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & 245x-630 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{-245x} & = & \color{red}{245x} -630 \color{blue}{-245x} \color{blue}{-28} \\\Leftrightarrow & -215x& = & -658 \\\Leftrightarrow & \frac{-215x}{ \color{red}{-215} }& = & \frac{-658}{-215} \\\Leftrightarrow & x = \frac{658}{215} & & \\ & V = \left\{ \frac{658}{215} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{36}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 36x+120 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-36x} & = & \color{red}{36x} +120 \color{blue}{-36x} \color{blue}{+16} \\\Leftrightarrow & -21x& = & 136 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{136}{-21} \\\Leftrightarrow & x = \frac{-136}{21} & & \\ & V = \left\{ \frac{-136}{21} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{7}{5}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{98}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & 98x+140 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-98x} & = & \color{red}{98x} +140 \color{blue}{-98x} \color{blue}{+40} \\\Leftrightarrow & -63x& = & 180 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{180}{-63} \\\Leftrightarrow & x = \frac{-20}{7} & & \\ & V = \left\{ \frac{-20}{7} \right\} & \\\end{align}\)
5. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{-16}{ \color{blue}{20} }x-\frac{140}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & -16x-140 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{+16x} & = & \color{red}{-16x} -140 \color{blue}{+16x} \color{blue}{-6} \\\Leftrightarrow & 21x& = & -146 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-146}{21} \\\Leftrightarrow & x = \frac{-146}{21} & & \\ & V = \left\{ \frac{-146}{21} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 28x-588 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-28x} & = & \color{red}{28x} -588 \color{blue}{-28x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -564 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-564}{-7} \\\Leftrightarrow & x = \frac{564}{7} & & \\ & V = \left\{ \frac{564}{7} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{99}{ \color{blue}{66} }x-\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-12& = & 99x-132 \\\Leftrightarrow & 22x \color{red}{-12} \color{blue}{+12} \color{blue}{-99x} & = & \color{red}{99x} -132 \color{blue}{-99x} \color{blue}{+12} \\\Leftrightarrow & -77x& = & -120 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-120}{-77} \\\Leftrightarrow & x = \frac{120}{77} & & \\ & V = \left\{ \frac{120}{77} \right\} & \\\end{align}\)
8. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{7}{5}x-6 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{154}{ \color{blue}{110} }x-\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-40& = & 154x-660 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-154x} & = & \color{red}{154x} -660 \color{blue}{-154x} \color{blue}{+40} \\\Leftrightarrow & -99x& = & -620 \\\Leftrightarrow & \frac{-99x}{ \color{red}{-99} }& = & \frac{-620}{-99} \\\Leftrightarrow & x = \frac{620}{99} & & \\ & V = \left\{ \frac{620}{99} \right\} & \\\end{align}\)
9. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{9}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }+ \frac{ 25 }{ \color{blue}{45} })& = & (\frac{-63}{ \color{blue}{45} }x+\frac{225}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x+25& = & -63x+225 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{+63x} & = & \color{red}{-63x} +225 \color{blue}{+63x} \color{blue}{-25} \\\Leftrightarrow & 78x& = & 200 \\\Leftrightarrow & \frac{78x}{ \color{red}{78} }& = & \frac{200}{78} \\\Leftrightarrow & x = \frac{100}{39} & & \\ & V = \left\{ \frac{100}{39} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 63x+252 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-63x} & = & \color{red}{63x} +252 \color{blue}{-63x} \color{blue}{-12} \\\Leftrightarrow & -49x& = & 240 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{240}{-49} \\\Leftrightarrow & x = \frac{-240}{49} & & \\ & V = \left\{ \frac{-240}{49} \right\} & \\\end{align}\)
11. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{11}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 60 }{ \color{blue}{220} })& = & (\frac{-308}{ \color{blue}{220} }x+\frac{1100}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-60& = & -308x+1100 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+308x} & = & \color{red}{-308x} +1100 \color{blue}{+308x} \color{blue}{+60} \\\Leftrightarrow & 363x& = & 1160 \\\Leftrightarrow & \frac{363x}{ \color{red}{363} }& = & \frac{1160}{363} \\\Leftrightarrow & x = \frac{1160}{363} & & \\ & V = \left\{ \frac{1160}{363} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -98x+350 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+98x} & = & \color{red}{-98x} +350 \color{blue}{+98x} \color{blue}{+20} \\\Leftrightarrow & 133x& = & 370 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{370}{133} \\\Leftrightarrow & x = \frac{370}{133} & & \\ & V = \left\{ \frac{370}{133} \right\} & \\\end{align}\)