Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{-182}{ \color{blue}{130} }x-\frac{520}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-30& = & -182x-520 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{+182x} & = & \color{red}{-182x} -520 \color{blue}{+182x} \color{blue}{+30} \\\Leftrightarrow & 247x& = & -490 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-490}{247} \\\Leftrightarrow & x = \frac{-490}{247} & & \\ & V = \left\{ \frac{-490}{247} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{5}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 105x+42 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-105x} & = & \color{red}{105x} +42 \color{blue}{-105x} \color{blue}{-24} \\\Leftrightarrow & -91x& = & 18 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{18}{-91} \\\Leftrightarrow & x = \frac{-18}{91} & & \\ & V = \left\{ \frac{-18}{91} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{5}{2}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{200}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 200x+400 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-200x} & = & \color{red}{200x} +400 \color{blue}{-200x} \color{blue}{-15} \\\Leftrightarrow & -184x& = & 385 \\\Leftrightarrow & \frac{-184x}{ \color{red}{-184} }& = & \frac{385}{-184} \\\Leftrightarrow & x = \frac{-385}{184} & & \\ & V = \left\{ \frac{-385}{184} \right\} & \\\end{align}\)
4. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{16}{ \color{blue}{20} }x+\frac{40}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & 16x+40 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{-16x} & = & \color{red}{16x} +40 \color{blue}{-16x} \color{blue}{-6} \\\Leftrightarrow & -11x& = & 34 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{34}{-11} \\\Leftrightarrow & x = \frac{-34}{11} & & \\ & V = \left\{ \frac{-34}{11} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x-\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & -546x-780 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+546x} & = & \color{red}{-546x} -780 \color{blue}{+546x} \color{blue}{-60} \\\Leftrightarrow & 611x& = & -840 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{-840}{611} \\\Leftrightarrow & x = \frac{-840}{611} & & \\ & V = \left\{ \frac{-840}{611} \right\} & \\\end{align}\)
6. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{3}{5}x+6 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }- \frac{ 45 }{ \color{blue}{195} })& = & (\frac{117}{ \color{blue}{195} }x+\frac{1170}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x-45& = & 117x+1170 \\\Leftrightarrow & 65x \color{red}{-45} \color{blue}{+45} \color{blue}{-117x} & = & \color{red}{117x} +1170 \color{blue}{-117x} \color{blue}{+45} \\\Leftrightarrow & -52x& = & 1215 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{1215}{-52} \\\Leftrightarrow & x = \frac{-1215}{52} & & \\ & V = \left\{ \frac{-1215}{52} \right\} & \\\end{align}\)
7. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x+\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -192x+1920 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+192x} & = & \color{red}{-192x} +1920 \color{blue}{+192x} \color{blue}{-75} \\\Leftrightarrow & 232x& = & 1845 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{1845}{232} \\\Leftrightarrow & x = \frac{1845}{232} & & \\ & V = \left\{ \frac{1845}{232} \right\} & \\\end{align}\)
8. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{5}{4}x-6 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{45}{ \color{blue}{36} }x-\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & 45x-216 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{-45x} & = & \color{red}{45x} -216 \color{blue}{-45x} \color{blue}{-8} \\\Leftrightarrow & -33x& = & -224 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-224}{-33} \\\Leftrightarrow & x = \frac{224}{33} & & \\ & V = \left\{ \frac{224}{33} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{4}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 112x-672 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-112x} & = & \color{red}{112x} -672 \color{blue}{-112x} \color{blue}{-36} \\\Leftrightarrow & -91x& = & -708 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-708}{-91} \\\Leftrightarrow & x = \frac{708}{91} & & \\ & V = \left\{ \frac{708}{91} \right\} & \\\end{align}\)
10. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 100 }{ \color{blue}{220} })& = & (\frac{44}{ \color{blue}{220} }x+\frac{220}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+100& = & 44x+220 \\\Leftrightarrow & 55x \color{red}{+100} \color{blue}{-100} \color{blue}{-44x} & = & \color{red}{44x} +220 \color{blue}{-44x} \color{blue}{-100} \\\Leftrightarrow & 11x& = & 120 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{120}{11} \\\Leftrightarrow & x = \frac{120}{11} & & \\ & V = \left\{ \frac{120}{11} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{126}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & 126x+540 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{-126x} & = & \color{red}{126x} +540 \color{blue}{-126x} \color{blue}{+50} \\\Leftrightarrow & -111x& = & 590 \\\Leftrightarrow & \frac{-111x}{ \color{red}{-111} }& = & \frac{590}{-111} \\\Leftrightarrow & x = \frac{-590}{111} & & \\ & V = \left\{ \frac{-590}{111} \right\} & \\\end{align}\)
12. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }- \frac{ 140 }{ \color{blue}{455} })& = & (\frac{-364}{ \color{blue}{455} }x-\frac{1365}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x-140& = & -364x-1365 \\\Leftrightarrow & 65x \color{red}{-140} \color{blue}{+140} \color{blue}{+364x} & = & \color{red}{-364x} -1365 \color{blue}{+364x} \color{blue}{+140} \\\Leftrightarrow & 429x& = & -1225 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-1225}{429} \\\Leftrightarrow & x = \frac{-1225}{429} & & \\ & V = \left\{ \frac{-1225}{429} \right\} & \\\end{align}\)