Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x+\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-60& = & 65x+520 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{-65x} & = & \color{red}{65x} +520 \color{blue}{-65x} \color{blue}{+60} \\\Leftrightarrow & -13x& = & 580 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{580}{-13} \\\Leftrightarrow & x = \frac{-580}{13} & & \\ & V = \left\{ \frac{-580}{13} \right\} & \\\end{align}\)
2. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{-154}{ \color{blue}{110} }x+\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+50& = & -154x+550 \\\Leftrightarrow & 55x \color{red}{+50} \color{blue}{-50} \color{blue}{+154x} & = & \color{red}{-154x} +550 \color{blue}{+154x} \color{blue}{-50} \\\Leftrightarrow & 209x& = & 500 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{500}{209} \\\Leftrightarrow & x = \frac{500}{209} & & \\ & V = \left\{ \frac{500}{209} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 30 }{ \color{blue}{165} })& = & (\frac{-440}{ \color{blue}{165} }x+\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+30& = & -440x+330 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+440x} & = & \color{red}{-440x} +330 \color{blue}{+440x} \color{blue}{-30} \\\Leftrightarrow & 473x& = & 300 \\\Leftrightarrow & \frac{473x}{ \color{red}{473} }& = & \frac{300}{473} \\\Leftrightarrow & x = \frac{300}{473} & & \\ & V = \left\{ \frac{300}{473} \right\} & \\\end{align}\)
4. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 42 }{ \color{blue}{231} })& = & (\frac{-154}{ \color{blue}{231} }x+\frac{1155}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+42& = & -154x+1155 \\\Leftrightarrow & 33x \color{red}{+42} \color{blue}{-42} \color{blue}{+154x} & = & \color{red}{-154x} +1155 \color{blue}{+154x} \color{blue}{-42} \\\Leftrightarrow & 187x& = & 1113 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{1113}{187} \\\Leftrightarrow & x = \frac{1113}{187} & & \\ & V = \left\{ \frac{1113}{187} \right\} & \\\end{align}\)
5. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{5}{3}x-8 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{10}{ \color{blue}{6} }x-\frac{48}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 10x-48 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-10x} & = & \color{red}{10x} -48 \color{blue}{-10x} \color{blue}{-5} \\\Leftrightarrow & -7x& = & -53 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-53}{-7} \\\Leftrightarrow & x = \frac{53}{7} & & \\ & V = \left\{ \frac{53}{7} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{-154}{ \color{blue}{110} }x-\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+40& = & -154x-550 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{+154x} & = & \color{red}{-154x} -550 \color{blue}{+154x} \color{blue}{-40} \\\Leftrightarrow & 209x& = & -590 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-590}{209} \\\Leftrightarrow & x = \frac{-590}{209} & & \\ & V = \left\{ \frac{-590}{209} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 80x+420 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-80x} & = & \color{red}{80x} +420 \color{blue}{-80x} \color{blue}{+18} \\\Leftrightarrow & -65x& = & 438 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{438}{-65} \\\Leftrightarrow & x = \frac{-438}{65} & & \\ & V = \left\{ \frac{-438}{65} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -72x-450 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+72x} & = & \color{red}{-72x} -450 \color{blue}{+72x} \color{blue}{-20} \\\Leftrightarrow & 87x& = & -470 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{-470}{87} \\\Leftrightarrow & x = \frac{-470}{87} & & \\ & V = \left\{ \frac{-470}{87} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 21x-252 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & -7x& = & -234 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-234}{-7} \\\Leftrightarrow & x = \frac{234}{7} & & \\ & V = \left\{ \frac{234}{7} \right\} & \\\end{align}\)
10. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{7}{5}x-1 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{336}{ \color{blue}{240} }x-\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 336x-240 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-336x} & = & \color{red}{336x} -240 \color{blue}{-336x} \color{blue}{+45} \\\Leftrightarrow & -296x& = & -195 \\\Leftrightarrow & \frac{-296x}{ \color{red}{-296} }& = & \frac{-195}{-296} \\\Leftrightarrow & x = \frac{195}{296} & & \\ & V = \left\{ \frac{195}{296} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{-7}{4}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & -147x+168 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{+147x} & = & \color{red}{-147x} +168 \color{blue}{+147x} \color{blue}{+35} \\\Leftrightarrow & 159x& = & 203 \\\Leftrightarrow & \frac{159x}{ \color{red}{159} }& = & \frac{203}{159} \\\Leftrightarrow & x = \frac{203}{159} & & \\ & V = \left\{ \frac{203}{159} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{14}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 30 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-30& = & -224x-672 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{+224x} & = & \color{red}{-224x} -672 \color{blue}{+224x} \color{blue}{+30} \\\Leftrightarrow & 245x& = & -642 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-642}{245} \\\Leftrightarrow & x = \frac{-642}{245} & & \\ & V = \left\{ \frac{-642}{245} \right\} & \\\end{align}\)