Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{5}{6}=\frac{7}{3}x+6\)
2. \(\frac{x}{5}-\frac{3}{10}=\frac{1}{6}x-3\)
3. \(\frac{x}{2}+\frac{4}{15}=\frac{1}{5}x-2\)
4. \(\frac{x}{2}+\frac{3}{16}=\frac{6}{5}x+4\)
5. \(\frac{x}{2}+\frac{5}{13}=\frac{1}{5}x+2\)
6. \(\frac{x}{6}-\frac{5}{6}=\frac{6}{5}x-4\)
7. \(\frac{x}{3}+\frac{2}{15}=\frac{3}{5}x+8\)
8. \(\frac{x}{4}-\frac{4}{15}=\frac{5}{3}x+6\)
9. \(\frac{x}{2}+\frac{5}{7}=\frac{1}{3}x-6\)
10. \(\frac{x}{5}+\frac{2}{15}=\frac{-3}{4}x+7\)
11. \(\frac{x}{6}+\frac{2}{11}=\frac{1}{5}x+1\)
12. \(\frac{x}{2}-\frac{4}{13}=\frac{1}{3}x-8\)

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

Verbetersleutel

1. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{7}{3}x+6 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{14}{ \color{blue}{6} }x+\frac{36}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 14x+36 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-14x} & = & \color{red}{14x} +36 \color{blue}{-14x} \color{blue}{-5} \\\Leftrightarrow & -11x& = & 31 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{31}{-11} \\\Leftrightarrow & x = \frac{-31}{11} & & \\ & V = \left\{ \frac{-31}{11} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & 5x-90 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-5x} & = & \color{red}{5x} -90 \color{blue}{-5x} \color{blue}{+9} \\\Leftrightarrow & x& = & -81 \\ & V = \left\{ -81 \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 6x-60 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-6x} & = & \color{red}{6x} -60 \color{blue}{-6x} \color{blue}{-8} \\\Leftrightarrow & 9x& = & -68 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{-68}{9} \\\Leftrightarrow & x = \frac{-68}{9} & & \\ & V = \left\{ \frac{-68}{9} \right\} & \\\end{align}\)
4. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{96}{ \color{blue}{80} }x+\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+15& = & 96x+320 \\\Leftrightarrow & 40x \color{red}{+15} \color{blue}{-15} \color{blue}{-96x} & = & \color{red}{96x} +320 \color{blue}{-96x} \color{blue}{-15} \\\Leftrightarrow & -56x& = & 305 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{305}{-56} \\\Leftrightarrow & x = \frac{-305}{56} & & \\ & V = \left\{ \frac{-305}{56} \right\} & \\\end{align}\)
5. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{13}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 50 }{ \color{blue}{130} })& = & (\frac{26}{ \color{blue}{130} }x+\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+50& = & 26x+260 \\\Leftrightarrow & 65x \color{red}{+50} \color{blue}{-50} \color{blue}{-26x} & = & \color{red}{26x} +260 \color{blue}{-26x} \color{blue}{-50} \\\Leftrightarrow & 39x& = & 210 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{210}{39} \\\Leftrightarrow & x = \frac{70}{13} & & \\ & V = \left\{ \frac{70}{13} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & 36x-120 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{-36x} & = & \color{red}{36x} -120 \color{blue}{-36x} \color{blue}{+25} \\\Leftrightarrow & -31x& = & -95 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-95}{-31} \\\Leftrightarrow & x = \frac{95}{31} & & \\ & V = \left\{ \frac{95}{31} \right\} & \\\end{align}\)
7. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{3}{5}x+8 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 2 }{ \color{blue}{15} })& = & (\frac{9}{ \color{blue}{15} }x+\frac{120}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+2& = & 9x+120 \\\Leftrightarrow & 5x \color{red}{+2} \color{blue}{-2} \color{blue}{-9x} & = & \color{red}{9x} +120 \color{blue}{-9x} \color{blue}{-2} \\\Leftrightarrow & -4x& = & 118 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{118}{-4} \\\Leftrightarrow & x = \frac{-59}{2} & & \\ & V = \left\{ \frac{-59}{2} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{5}{3}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{100}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 100x+360 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-100x} & = & \color{red}{100x} +360 \color{blue}{-100x} \color{blue}{+16} \\\Leftrightarrow & -85x& = & 376 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{376}{-85} \\\Leftrightarrow & x = \frac{-376}{85} & & \\ & V = \left\{ \frac{-376}{85} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 30 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+30& = & 14x-252 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-14x} & = & \color{red}{14x} -252 \color{blue}{-14x} \color{blue}{-30} \\\Leftrightarrow & 7x& = & -282 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-282}{7} \\\Leftrightarrow & x = \frac{-282}{7} & & \\ & V = \left\{ \frac{-282}{7} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+8& = & -45x+420 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{+45x} & = & \color{red}{-45x} +420 \color{blue}{+45x} \color{blue}{-8} \\\Leftrightarrow & 57x& = & 412 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{412}{57} \\\Leftrightarrow & x = \frac{412}{57} & & \\ & V = \left\{ \frac{412}{57} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{11}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & 66x+330 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-66x} & = & \color{red}{66x} +330 \color{blue}{-66x} \color{blue}{-60} \\\Leftrightarrow & -11x& = & 270 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{270}{-11} \\\Leftrightarrow & x = \frac{-270}{11} & & \\ & V = \left\{ \frac{-270}{11} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{1}{3}x-8 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & 26x-624 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-26x} & = & \color{red}{26x} -624 \color{blue}{-26x} \color{blue}{+24} \\\Leftrightarrow & 13x& = & -600 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-600}{13} \\\Leftrightarrow & x = \frac{-600}{13} & & \\ & V = \left\{ \frac{-600}{13} \right\} & \\\end{align}\)