Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 42 }{ \color{blue}{182} })& = & (\frac{637}{ \color{blue}{182} }x-\frac{546}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+42& = & 637x-546 \\\Leftrightarrow & 26x \color{red}{+42} \color{blue}{-42} \color{blue}{-637x} & = & \color{red}{637x} -546 \color{blue}{-637x} \color{blue}{-42} \\\Leftrightarrow & -611x& = & -588 \\\Leftrightarrow & \frac{-611x}{ \color{red}{-611} }& = & \frac{-588}{-611} \\\Leftrightarrow & x = \frac{588}{611} & & \\ & V = \left\{ \frac{588}{611} \right\} & \\\end{align}\)
2. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{5}{4}x+4 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{325}{ \color{blue}{260} }x+\frac{1040}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-80& = & 325x+1040 \\\Leftrightarrow & 52x \color{red}{-80} \color{blue}{+80} \color{blue}{-325x} & = & \color{red}{325x} +1040 \color{blue}{-325x} \color{blue}{+80} \\\Leftrightarrow & -273x& = & 1120 \\\Leftrightarrow & \frac{-273x}{ \color{red}{-273} }& = & \frac{1120}{-273} \\\Leftrightarrow & x = \frac{-160}{39} & & \\ & V = \left\{ \frac{-160}{39} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -70x+252 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+70x} & = & \color{red}{-70x} +252 \color{blue}{+70x} \color{blue}{-12} \\\Leftrightarrow & 91x& = & 240 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{240}{91} \\\Leftrightarrow & x = \frac{240}{91} & & \\ & V = \left\{ \frac{240}{91} \right\} & \\\end{align}\)
4. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{5}{3}x+5 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 42 }{ \color{blue}{273} })& = & (\frac{455}{ \color{blue}{273} }x+\frac{1365}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-42& = & 455x+1365 \\\Leftrightarrow & 39x \color{red}{-42} \color{blue}{+42} \color{blue}{-455x} & = & \color{red}{455x} +1365 \color{blue}{-455x} \color{blue}{+42} \\\Leftrightarrow & -416x& = & 1407 \\\Leftrightarrow & \frac{-416x}{ \color{red}{-416} }& = & \frac{1407}{-416} \\\Leftrightarrow & x = \frac{-1407}{416} & & \\ & V = \left\{ \frac{-1407}{416} \right\} & \\\end{align}\)
5. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{-154}{ \color{blue}{110} }x-\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-50& = & -154x-440 \\\Leftrightarrow & 55x \color{red}{-50} \color{blue}{+50} \color{blue}{+154x} & = & \color{red}{-154x} -440 \color{blue}{+154x} \color{blue}{+50} \\\Leftrightarrow & 209x& = & -390 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-390}{209} \\\Leftrightarrow & x = \frac{-390}{209} & & \\ & V = \left\{ \frac{-390}{209} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x+\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x-45& = & 48x+1680 \\\Leftrightarrow & 80x \color{red}{-45} \color{blue}{+45} \color{blue}{-48x} & = & \color{red}{48x} +1680 \color{blue}{-48x} \color{blue}{+45} \\\Leftrightarrow & 32x& = & 1725 \\\Leftrightarrow & \frac{32x}{ \color{red}{32} }& = & \frac{1725}{32} \\\Leftrightarrow & x = \frac{1725}{32} & & \\ & V = \left\{ \frac{1725}{32} \right\} & \\\end{align}\)
7. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{1040}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-30& = & 65x-1040 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-65x} & = & \color{red}{65x} -1040 \color{blue}{-65x} \color{blue}{+30} \\\Leftrightarrow & -39x& = & -1010 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-1010}{-39} \\\Leftrightarrow & x = \frac{1010}{39} & & \\ & V = \left\{ \frac{1010}{39} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{-3}{4}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-24& = & -117x+624 \\\Leftrightarrow & 52x \color{red}{-24} \color{blue}{+24} \color{blue}{+117x} & = & \color{red}{-117x} +624 \color{blue}{+117x} \color{blue}{+24} \\\Leftrightarrow & 169x& = & 648 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{648}{169} \\\Leftrightarrow & x = \frac{648}{169} & & \\ & V = \left\{ \frac{648}{169} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x-\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & -80x-384 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{+80x} & = & \color{red}{-80x} -384 \color{blue}{+80x} \color{blue}{-15} \\\Leftrightarrow & 104x& = & -399 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{-399}{104} \\\Leftrightarrow & x = \frac{-399}{104} & & \\ & V = \left\{ \frac{-399}{104} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{2}{5}x-3 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{52}{ \color{blue}{130} }x-\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+30& = & 52x-390 \\\Leftrightarrow & 65x \color{red}{+30} \color{blue}{-30} \color{blue}{-52x} & = & \color{red}{52x} -390 \color{blue}{-52x} \color{blue}{-30} \\\Leftrightarrow & 13x& = & -420 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-420}{13} \\\Leftrightarrow & x = \frac{-420}{13} & & \\ & V = \left\{ \frac{-420}{13} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-24& = & 39x-234 \\\Leftrightarrow & 26x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} -234 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & -210 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-210}{-13} \\\Leftrightarrow & x = \frac{210}{13} & & \\ & V = \left\{ \frac{210}{13} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+50& = & 55x+110 \\\Leftrightarrow & 22x \color{red}{+50} \color{blue}{-50} \color{blue}{-55x} & = & \color{red}{55x} +110 \color{blue}{-55x} \color{blue}{-50} \\\Leftrightarrow & -33x& = & 60 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{60}{-33} \\\Leftrightarrow & x = \frac{-20}{11} & & \\ & V = \left\{ \frac{-20}{11} \right\} & \\\end{align}\)