Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}+\frac{5}{9}=\frac{1}{2}x-2\)
2. \(\frac{x}{5}-\frac{2}{15}=\frac{1}{2}x-2\)
3. \(\frac{x}{6}+\frac{2}{7}=\frac{2}{5}x-5\)
4. \(\frac{x}{5}-\frac{5}{14}=\frac{7}{3}x-8\)
5. \(\frac{x}{7}-\frac{4}{13}=\frac{1}{2}x-4\)
6. \(\frac{x}{2}+\frac{5}{9}=\frac{7}{3}x-4\)
7. \(\frac{x}{2}-\frac{2}{9}=\frac{-7}{5}x+4\)
8. \(\frac{x}{6}-\frac{5}{6}=\frac{3}{5}x+5\)
9. \(\frac{x}{3}-\frac{2}{7}=\frac{3}{2}x-3\)
10. \(\frac{x}{6}-\frac{4}{9}=\frac{-2}{5}x-7\)
11. \(\frac{x}{4}+\frac{3}{7}=\frac{7}{3}x-8\)
12. \(\frac{x}{3}-\frac{3}{7}=\frac{1}{2}x-4\)

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

Verbetersleutel

1. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{9}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+10& = & 9x-36 \\\Leftrightarrow & 6x \color{red}{+10} \color{blue}{-10} \color{blue}{-9x} & = & \color{red}{9x} -36 \color{blue}{-9x} \color{blue}{-10} \\\Leftrightarrow & -3x& = & -46 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-46}{-3} \\\Leftrightarrow & x = \frac{46}{3} & & \\ & V = \left\{ \frac{46}{3} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 15x-60 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} -60 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -9x& = & -56 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-56}{-9} \\\Leftrightarrow & x = \frac{56}{9} & & \\ & V = \left\{ \frac{56}{9} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{2}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 84x-1050 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-84x} & = & \color{red}{84x} -1050 \color{blue}{-84x} \color{blue}{-60} \\\Leftrightarrow & -49x& = & -1110 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-1110}{-49} \\\Leftrightarrow & x = \frac{1110}{49} & & \\ & V = \left\{ \frac{1110}{49} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{14}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{490}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-75& = & 490x-1680 \\\Leftrightarrow & 42x \color{red}{-75} \color{blue}{+75} \color{blue}{-490x} & = & \color{red}{490x} -1680 \color{blue}{-490x} \color{blue}{+75} \\\Leftrightarrow & -448x& = & -1605 \\\Leftrightarrow & \frac{-448x}{ \color{red}{-448} }& = & \frac{-1605}{-448} \\\Leftrightarrow & x = \frac{1605}{448} & & \\ & V = \left\{ \frac{1605}{448} \right\} & \\\end{align}\)
5. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 56 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{728}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-56& = & 91x-728 \\\Leftrightarrow & 26x \color{red}{-56} \color{blue}{+56} \color{blue}{-91x} & = & \color{red}{91x} -728 \color{blue}{-91x} \color{blue}{+56} \\\Leftrightarrow & -65x& = & -672 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-672}{-65} \\\Leftrightarrow & x = \frac{672}{65} & & \\ & V = \left\{ \frac{672}{65} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{42}{ \color{blue}{18} }x-\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+10& = & 42x-72 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{-42x} & = & \color{red}{42x} -72 \color{blue}{-42x} \color{blue}{-10} \\\Leftrightarrow & -33x& = & -82 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-82}{-33} \\\Leftrightarrow & x = \frac{82}{33} & & \\ & V = \left\{ \frac{82}{33} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & -126x+360 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{+126x} & = & \color{red}{-126x} +360 \color{blue}{+126x} \color{blue}{+20} \\\Leftrightarrow & 171x& = & 380 \\\Leftrightarrow & \frac{171x}{ \color{red}{171} }& = & \frac{380}{171} \\\Leftrightarrow & x = \frac{20}{9} & & \\ & V = \left\{ \frac{20}{9} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{3}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & 18x+150 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{-18x} & = & \color{red}{18x} +150 \color{blue}{-18x} \color{blue}{+25} \\\Leftrightarrow & -13x& = & 175 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{175}{-13} \\\Leftrightarrow & x = \frac{-175}{13} & & \\ & V = \left\{ \frac{-175}{13} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 63x-126 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-63x} & = & \color{red}{63x} -126 \color{blue}{-63x} \color{blue}{+12} \\\Leftrightarrow & -49x& = & -114 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-114}{-49} \\\Leftrightarrow & x = \frac{114}{49} & & \\ & V = \left\{ \frac{114}{49} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -36x-630 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+36x} & = & \color{red}{-36x} -630 \color{blue}{+36x} \color{blue}{+40} \\\Leftrightarrow & 51x& = & -590 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-590}{51} \\\Leftrightarrow & x = \frac{-590}{51} & & \\ & V = \left\{ \frac{-590}{51} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 196x-672 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-196x} & = & \color{red}{196x} -672 \color{blue}{-196x} \color{blue}{-36} \\\Leftrightarrow & -175x& = & -708 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-708}{-175} \\\Leftrightarrow & x = \frac{708}{175} & & \\ & V = \left\{ \frac{708}{175} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 21x-168 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & -7x& = & -150 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-150}{-7} \\\Leftrightarrow & x = \frac{150}{7} & & \\ & V = \left\{ \frac{150}{7} \right\} & \\\end{align}\)