Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{7}{4}x-5 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{63}{ \color{blue}{36} }x-\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-20& = & 63x-180 \\\Leftrightarrow & 12x \color{red}{-20} \color{blue}{+20} \color{blue}{-63x} & = & \color{red}{63x} -180 \color{blue}{-63x} \color{blue}{+20} \\\Leftrightarrow & -51x& = & -160 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{-160}{-51} \\\Leftrightarrow & x = \frac{160}{51} & & \\ & V = \left\{ \frac{160}{51} \right\} & \\\end{align}\)
2. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{11}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 63 }{ \color{blue}{231} })& = & (\frac{-154}{ \color{blue}{231} }x-\frac{924}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+63& = & -154x-924 \\\Leftrightarrow & 33x \color{red}{+63} \color{blue}{-63} \color{blue}{+154x} & = & \color{red}{-154x} -924 \color{blue}{+154x} \color{blue}{-63} \\\Leftrightarrow & 187x& = & -987 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-987}{187} \\\Leftrightarrow & x = \frac{-987}{187} & & \\ & V = \left\{ \frac{-987}{187} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x+126 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} +126 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 150 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{150}{-7} \\\Leftrightarrow & x = \frac{-150}{7} & & \\ & V = \left\{ \frac{-150}{7} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{1}{6}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 5x-150 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-5x} & = & \color{red}{5x} -150 \color{blue}{-5x} \color{blue}{-25} \\\Leftrightarrow & x& = & -175 \\ & V = \left\{ -175 \right\} & \\\end{align}\)
5. \(\text{40 is het kleinste gemene veelvoud van 4, 8 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{8}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{40.} (\frac{10x}{ \color{blue}{40} }- \frac{ 25 }{ \color{blue}{40} })& = & (\frac{-56}{ \color{blue}{40} }x-\frac{120}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 10x-25& = & -56x-120 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{+56x} & = & \color{red}{-56x} -120 \color{blue}{+56x} \color{blue}{+25} \\\Leftrightarrow & 66x& = & -95 \\\Leftrightarrow & \frac{66x}{ \color{red}{66} }& = & \frac{-95}{66} \\\Leftrightarrow & x = \frac{-95}{66} & & \\ & V = \left\{ \frac{-95}{66} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & -56x+504 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{+56x} & = & \color{red}{-56x} +504 \color{blue}{+56x} \color{blue}{+36} \\\Leftrightarrow & 77x& = & 540 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{540}{77} \\\Leftrightarrow & x = \frac{540}{77} & & \\ & V = \left\{ \frac{540}{77} \right\} & \\\end{align}\)
7. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-96}{ \color{blue}{120} }x+\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & -96x+360 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{+96x} & = & \color{red}{-96x} +360 \color{blue}{+96x} \color{blue}{-75} \\\Leftrightarrow & 116x& = & 285 \\\Leftrightarrow & \frac{116x}{ \color{red}{116} }& = & \frac{285}{116} \\\Leftrightarrow & x = \frac{285}{116} & & \\ & V = \left\{ \frac{285}{116} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{14}& = & \frac{5}{6}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 15 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-15& = & 35x+42 \\\Leftrightarrow & 6x \color{red}{-15} \color{blue}{+15} \color{blue}{-35x} & = & \color{red}{35x} +42 \color{blue}{-35x} \color{blue}{+15} \\\Leftrightarrow & -29x& = & 57 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{57}{-29} \\\Leftrightarrow & x = \frac{-57}{29} & & \\ & V = \left\{ \frac{-57}{29} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x-\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & -104x-936 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{+104x} & = & \color{red}{-104x} -936 \color{blue}{+104x} \color{blue}{+60} \\\Leftrightarrow & 143x& = & -876 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-876}{143} \\\Leftrightarrow & x = \frac{-876}{143} & & \\ & V = \left\{ \frac{-876}{143} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x+650 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} +650 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & 620 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{620}{-39} \\\Leftrightarrow & x = \frac{-620}{39} & & \\ & V = \left\{ \frac{-620}{39} \right\} & \\\end{align}\)
11. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{-44}{ \color{blue}{110} }x+\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-50& = & -44x+550 \\\Leftrightarrow & 55x \color{red}{-50} \color{blue}{+50} \color{blue}{+44x} & = & \color{red}{-44x} +550 \color{blue}{+44x} \color{blue}{+50} \\\Leftrightarrow & 99x& = & 600 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{600}{99} \\\Leftrightarrow & x = \frac{200}{33} & & \\ & V = \left\{ \frac{200}{33} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{1}{4}x-4 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 36 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x-\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-36& = & 33x-528 \\\Leftrightarrow & 44x \color{red}{-36} \color{blue}{+36} \color{blue}{-33x} & = & \color{red}{33x} -528 \color{blue}{-33x} \color{blue}{+36} \\\Leftrightarrow & 11x& = & -492 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-492}{11} \\\Leftrightarrow & x = \frac{-492}{11} & & \\ & V = \left\{ \frac{-492}{11} \right\} & \\\end{align}\)