Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 70 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{308}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-70& = & 77x-308 \\\Leftrightarrow & 22x \color{red}{-70} \color{blue}{+70} \color{blue}{-77x} & = & \color{red}{77x} -308 \color{blue}{-77x} \color{blue}{+70} \\\Leftrightarrow & -55x& = & -238 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-238}{-55} \\\Leftrightarrow & x = \frac{238}{55} & & \\ & V = \left\{ \frac{238}{55} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+8& = & 7x+14 \\\Leftrightarrow & 2x \color{red}{+8} \color{blue}{-8} \color{blue}{-7x} & = & \color{red}{7x} +14 \color{blue}{-7x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & 6 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{6}{-5} \\\Leftrightarrow & x = \frac{-6}{5} & & \\ & V = \left\{ \frac{-6}{5} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x+\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x+45& = & 48x+1440 \\\Leftrightarrow & 80x \color{red}{+45} \color{blue}{-45} \color{blue}{-48x} & = & \color{red}{48x} +1440 \color{blue}{-48x} \color{blue}{-45} \\\Leftrightarrow & 32x& = & 1395 \\\Leftrightarrow & \frac{32x}{ \color{red}{32} }& = & \frac{1395}{32} \\\Leftrightarrow & x = \frac{1395}{32} & & \\ & V = \left\{ \frac{1395}{32} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{5}{3}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{50}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 50x-30 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-50x} & = & \color{red}{50x} -30 \color{blue}{-50x} \color{blue}{-8} \\\Leftrightarrow & -35x& = & -38 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-38}{-35} \\\Leftrightarrow & x = \frac{38}{35} & & \\ & V = \left\{ \frac{38}{35} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{5}{3}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 63 }{ \color{blue}{210} })& = & (\frac{350}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-63& = & 350x+420 \\\Leftrightarrow & 30x \color{red}{-63} \color{blue}{+63} \color{blue}{-350x} & = & \color{red}{350x} +420 \color{blue}{-350x} \color{blue}{+63} \\\Leftrightarrow & -320x& = & 483 \\\Leftrightarrow & \frac{-320x}{ \color{red}{-320} }& = & \frac{483}{-320} \\\Leftrightarrow & x = \frac{-483}{320} & & \\ & V = \left\{ \frac{-483}{320} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -36x-720 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+36x} & = & \color{red}{-36x} -720 \color{blue}{+36x} \color{blue}{+40} \\\Leftrightarrow & 51x& = & -680 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-680}{51} \\\Leftrightarrow & x = \frac{-40}{3} & & \\ & V = \left\{ \frac{-40}{3} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x-\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-24& = & -99x-792 \\\Leftrightarrow & 44x \color{red}{-24} \color{blue}{+24} \color{blue}{+99x} & = & \color{red}{-99x} -792 \color{blue}{+99x} \color{blue}{+24} \\\Leftrightarrow & 143x& = & -768 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-768}{143} \\\Leftrightarrow & x = \frac{-768}{143} & & \\ & V = \left\{ \frac{-768}{143} \right\} & \\\end{align}\)
8. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{13}& = & \frac{5}{4}x+5 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 140 }{ \color{blue}{364} })& = & (\frac{455}{ \color{blue}{364} }x+\frac{1820}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-140& = & 455x+1820 \\\Leftrightarrow & 52x \color{red}{-140} \color{blue}{+140} \color{blue}{-455x} & = & \color{red}{455x} +1820 \color{blue}{-455x} \color{blue}{+140} \\\Leftrightarrow & -403x& = & 1960 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{1960}{-403} \\\Leftrightarrow & x = \frac{-1960}{403} & & \\ & V = \left\{ \frac{-1960}{403} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & -24x-90 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+24x} & = & \color{red}{-24x} -90 \color{blue}{+24x} \color{blue}{+25} \\\Leftrightarrow & 29x& = & -65 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-65}{29} \\\Leftrightarrow & x = \frac{-65}{29} & & \\ & V = \left\{ \frac{-65}{29} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{13}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 50 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x+\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+50& = & 65x+780 \\\Leftrightarrow & 26x \color{red}{+50} \color{blue}{-50} \color{blue}{-65x} & = & \color{red}{65x} +780 \color{blue}{-65x} \color{blue}{-50} \\\Leftrightarrow & -39x& = & 730 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{730}{-39} \\\Leftrightarrow & x = \frac{-730}{39} & & \\ & V = \left\{ \frac{-730}{39} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{4}{5}x-5 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{72}{ \color{blue}{90} }x-\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & 72x-450 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{-72x} & = & \color{red}{72x} -450 \color{blue}{-72x} \color{blue}{-20} \\\Leftrightarrow & -57x& = & -470 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{-470}{-57} \\\Leftrightarrow & x = \frac{470}{57} & & \\ & V = \left\{ \frac{470}{57} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -112x+252 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+112x} & = & \color{red}{-112x} +252 \color{blue}{+112x} \color{blue}{-12} \\\Leftrightarrow & 133x& = & 240 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{240}{133} \\\Leftrightarrow & x = \frac{240}{133} & & \\ & V = \left\{ \frac{240}{133} \right\} & \\\end{align}\)