Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}-\frac{5}{11}=\frac{-2}{3}x-8\)
2. \(\frac{x}{4}-\frac{4}{11}=\frac{1}{5}x+6\)
3. \(\frac{x}{7}-\frac{3}{13}=\frac{7}{2}x-4\)
4. \(\frac{x}{3}+\frac{5}{13}=\frac{1}{2}x+1\)
5. \(\frac{x}{5}-\frac{4}{11}=\frac{3}{2}x-3\)
6. \(\frac{x}{7}+\frac{5}{16}=\frac{1}{2}x-5\)
7. \(\frac{x}{4}-\frac{5}{16}=\frac{-7}{5}x+8\)
8. \(\frac{x}{5}+\frac{3}{10}=\frac{2}{3}x-8\)
9. \(\frac{x}{4}+\frac{5}{14}=\frac{2}{3}x-1\)
10. \(\frac{x}{7}-\frac{3}{7}=\frac{1}{2}x+6\)
11. \(\frac{x}{4}-\frac{2}{7}=\frac{-5}{3}x-3\)
12. \(\frac{x}{3}-\frac{3}{14}=\frac{-7}{4}x-6\)

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & -44x-528 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{+44x} & = & \color{red}{-44x} -528 \color{blue}{+44x} \color{blue}{+30} \\\Leftrightarrow & 77x& = & -498 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-498}{77} \\\Leftrightarrow & x = \frac{-498}{77} & & \\ & V = \left\{ \frac{-498}{77} \right\} & \\\end{align}\)
2. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 80 }{ \color{blue}{220} })& = & (\frac{44}{ \color{blue}{220} }x+\frac{1320}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-80& = & 44x+1320 \\\Leftrightarrow & 55x \color{red}{-80} \color{blue}{+80} \color{blue}{-44x} & = & \color{red}{44x} +1320 \color{blue}{-44x} \color{blue}{+80} \\\Leftrightarrow & 11x& = & 1400 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{1400}{11} \\\Leftrightarrow & x = \frac{1400}{11} & & \\ & V = \left\{ \frac{1400}{11} \right\} & \\\end{align}\)
3. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{7}{2}x-4 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{637}{ \color{blue}{182} }x-\frac{728}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 637x-728 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-637x} & = & \color{red}{637x} -728 \color{blue}{-637x} \color{blue}{+42} \\\Leftrightarrow & -611x& = & -686 \\\Leftrightarrow & \frac{-611x}{ \color{red}{-611} }& = & \frac{-686}{-611} \\\Leftrightarrow & x = \frac{686}{611} & & \\ & V = \left\{ \frac{686}{611} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{13}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 30 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+30& = & 39x+78 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-39x} & = & \color{red}{39x} +78 \color{blue}{-39x} \color{blue}{-30} \\\Leftrightarrow & -13x& = & 48 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{48}{-13} \\\Leftrightarrow & x = \frac{-48}{13} & & \\ & V = \left\{ \frac{-48}{13} \right\} & \\\end{align}\)
5. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{165}{ \color{blue}{110} }x-\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 165x-330 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-165x} & = & \color{red}{165x} -330 \color{blue}{-165x} \color{blue}{+40} \\\Leftrightarrow & -143x& = & -290 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-290}{-143} \\\Leftrightarrow & x = \frac{290}{143} & & \\ & V = \left\{ \frac{290}{143} \right\} & \\\end{align}\)
6. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{560}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 56x-560 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-56x} & = & \color{red}{56x} -560 \color{blue}{-56x} \color{blue}{-35} \\\Leftrightarrow & -40x& = & -595 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-595}{-40} \\\Leftrightarrow & x = \frac{119}{8} & & \\ & V = \left\{ \frac{119}{8} \right\} & \\\end{align}\)
7. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x+\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & -112x+640 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{+112x} & = & \color{red}{-112x} +640 \color{blue}{+112x} \color{blue}{+25} \\\Leftrightarrow & 132x& = & 665 \\\Leftrightarrow & \frac{132x}{ \color{red}{132} }& = & \frac{665}{132} \\\Leftrightarrow & x = \frac{665}{132} & & \\ & V = \left\{ \frac{665}{132} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{2}{3}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{20}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+9& = & 20x-240 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{-20x} & = & \color{red}{20x} -240 \color{blue}{-20x} \color{blue}{-9} \\\Leftrightarrow & -14x& = & -249 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{-249}{-14} \\\Leftrightarrow & x = \frac{249}{14} & & \\ & V = \left\{ \frac{249}{14} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{14}& = & \frac{2}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 30 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+30& = & 56x-84 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-56x} & = & \color{red}{56x} -84 \color{blue}{-56x} \color{blue}{-30} \\\Leftrightarrow & -35x& = & -114 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-114}{-35} \\\Leftrightarrow & x = \frac{114}{35} & & \\ & V = \left\{ \frac{114}{35} \right\} & \\\end{align}\)
10. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x+84 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} +84 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & 90 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{90}{-5} \\\Leftrightarrow & x = -18 & & \\ & V = \left\{ -18 \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -140x-252 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+140x} & = & \color{red}{-140x} -252 \color{blue}{+140x} \color{blue}{+24} \\\Leftrightarrow & 161x& = & -228 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-228}{161} \\\Leftrightarrow & x = \frac{-228}{161} & & \\ & V = \left\{ \frac{-228}{161} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{-7}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-18& = & -147x-504 \\\Leftrightarrow & 28x \color{red}{-18} \color{blue}{+18} \color{blue}{+147x} & = & \color{red}{-147x} -504 \color{blue}{+147x} \color{blue}{+18} \\\Leftrightarrow & 175x& = & -486 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-486}{175} \\\Leftrightarrow & x = \frac{-486}{175} & & \\ & V = \left\{ \frac{-486}{175} \right\} & \\\end{align}\)