Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 63 }{ \color{blue}{336} })& = & (\frac{-896}{ \color{blue}{336} }x+\frac{1680}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+63& = & -896x+1680 \\\Leftrightarrow & 48x \color{red}{+63} \color{blue}{-63} \color{blue}{+896x} & = & \color{red}{-896x} +1680 \color{blue}{+896x} \color{blue}{-63} \\\Leftrightarrow & 944x& = & 1617 \\\Leftrightarrow & \frac{944x}{ \color{red}{944} }& = & \frac{1617}{944} \\\Leftrightarrow & x = \frac{1617}{944} & & \\ & V = \left\{ \frac{1617}{944} \right\} & \\\end{align}\)
2. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{3}{5}x+1 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{6}{ \color{blue}{10} }x+\frac{10}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x+3& = & 6x+10 \\\Leftrightarrow & 5x \color{red}{+3} \color{blue}{-3} \color{blue}{-6x} & = & \color{red}{6x} +10 \color{blue}{-6x} \color{blue}{-3} \\\Leftrightarrow & -x& = & 7 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{7}{-1} \\\Leftrightarrow & x = -7 & & \\ & V = \left\{ -7 \right\} & \\\end{align}\)
3. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{364}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 91x-364 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-91x} & = & \color{red}{91x} -364 \color{blue}{-91x} \color{blue}{+42} \\\Leftrightarrow & -65x& = & -322 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-322}{-65} \\\Leftrightarrow & x = \frac{322}{65} & & \\ & V = \left\{ \frac{322}{65} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{-3}{4}x+6 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }+ \frac{ 100 }{ \color{blue}{220} })& = & (\frac{-165}{ \color{blue}{220} }x+\frac{1320}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x+100& = & -165x+1320 \\\Leftrightarrow & 44x \color{red}{+100} \color{blue}{-100} \color{blue}{+165x} & = & \color{red}{-165x} +1320 \color{blue}{+165x} \color{blue}{-100} \\\Leftrightarrow & 209x& = & 1220 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{1220}{209} \\\Leftrightarrow & x = \frac{1220}{209} & & \\ & V = \left\{ \frac{1220}{209} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & -45x-300 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{+45x} & = & \color{red}{-45x} -300 \color{blue}{+45x} \color{blue}{-25} \\\Leftrightarrow & 57x& = & -325 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-325}{57} \\\Leftrightarrow & x = \frac{-325}{57} & & \\ & V = \left\{ \frac{-325}{57} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & 18x-720 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{-18x} & = & \color{red}{18x} -720 \color{blue}{-18x} \color{blue}{+50} \\\Leftrightarrow & -3x& = & -670 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-670}{-3} \\\Leftrightarrow & x = \frac{670}{3} & & \\ & V = \left\{ \frac{670}{3} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{5}{3}x-1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{30}{ \color{blue}{18} }x-\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & 30x-18 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{-30x} & = & \color{red}{30x} -18 \color{blue}{-30x} \color{blue}{+4} \\\Leftrightarrow & -21x& = & -14 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-14}{-21} \\\Leftrightarrow & x = \frac{2}{3} & & \\ & V = \left\{ \frac{2}{3} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & -140x+336 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+140x} & = & \color{red}{-140x} +336 \color{blue}{+140x} \color{blue}{+18} \\\Leftrightarrow & 161x& = & 354 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{354}{161} \\\Leftrightarrow & x = \frac{354}{161} & & \\ & V = \left\{ \frac{354}{161} \right\} & \\\end{align}\)
9. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{-308}{ \color{blue}{220} }x+\frac{1760}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-40& = & -308x+1760 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{+308x} & = & \color{red}{-308x} +1760 \color{blue}{+308x} \color{blue}{+40} \\\Leftrightarrow & 363x& = & 1800 \\\Leftrightarrow & \frac{363x}{ \color{red}{363} }& = & \frac{1800}{363} \\\Leftrightarrow & x = \frac{600}{121} & & \\ & V = \left\{ \frac{600}{121} \right\} & \\\end{align}\)
10. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{11}& = & \frac{2}{3}x-8 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 63 }{ \color{blue}{231} })& = & (\frac{154}{ \color{blue}{231} }x-\frac{1848}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+63& = & 154x-1848 \\\Leftrightarrow & 33x \color{red}{+63} \color{blue}{-63} \color{blue}{-154x} & = & \color{red}{154x} -1848 \color{blue}{-154x} \color{blue}{-63} \\\Leftrightarrow & -121x& = & -1911 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-1911}{-121} \\\Leftrightarrow & x = \frac{1911}{121} & & \\ & V = \left\{ \frac{1911}{121} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & -44x-66 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+44x} & = & \color{red}{-44x} -66 \color{blue}{+44x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & -42 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-42}{77} \\\Leftrightarrow & x = \frac{-6}{11} & & \\ & V = \left\{ \frac{-6}{11} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{5}{3}x+3 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{80}{ \color{blue}{48} }x+\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & 80x+144 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{-80x} & = & \color{red}{80x} +144 \color{blue}{-80x} \color{blue}{+9} \\\Leftrightarrow & -56x& = & 153 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{153}{-56} \\\Leftrightarrow & x = \frac{-153}{56} & & \\ & V = \left\{ \frac{-153}{56} \right\} & \\\end{align}\)