Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+24& = & 88x+66 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-88x} & = & \color{red}{88x} +66 \color{blue}{-88x} \color{blue}{-24} \\\Leftrightarrow & -55x& = & 42 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{42}{-55} \\\Leftrightarrow & x = \frac{-42}{55} & & \\ & V = \left\{ \frac{-42}{55} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-40& = & 65x-130 \\\Leftrightarrow & 26x \color{red}{-40} \color{blue}{+40} \color{blue}{-65x} & = & \color{red}{65x} -130 \color{blue}{-65x} \color{blue}{+40} \\\Leftrightarrow & -39x& = & -90 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-90}{-39} \\\Leftrightarrow & x = \frac{30}{13} & & \\ & V = \left\{ \frac{30}{13} \right\} & \\\end{align}\)
3. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x-\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & 176x-660 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-176x} & = & \color{red}{176x} -660 \color{blue}{-176x} \color{blue}{-60} \\\Leftrightarrow & -143x& = & -720 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-720}{-143} \\\Leftrightarrow & x = \frac{720}{143} & & \\ & V = \left\{ \frac{720}{143} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{2}{3}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & 56x-504 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-56x} & = & \color{red}{56x} -504 \color{blue}{-56x} \color{blue}{+18} \\\Leftrightarrow & -35x& = & -486 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-486}{-35} \\\Leftrightarrow & x = \frac{486}{35} & & \\ & V = \left\{ \frac{486}{35} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-16& = & -45x+180 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{+45x} & = & \color{red}{-45x} +180 \color{blue}{+45x} \color{blue}{+16} \\\Leftrightarrow & 57x& = & 196 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{196}{57} \\\Leftrightarrow & x = \frac{196}{57} & & \\ & V = \left\{ \frac{196}{57} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{3}{4}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{45}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+16& = & 45x-60 \\\Leftrightarrow & 20x \color{red}{+16} \color{blue}{-16} \color{blue}{-45x} & = & \color{red}{45x} -60 \color{blue}{-45x} \color{blue}{-16} \\\Leftrightarrow & -25x& = & -76 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-76}{-25} \\\Leftrightarrow & x = \frac{76}{25} & & \\ & V = \left\{ \frac{76}{25} \right\} & \\\end{align}\)
7. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 140 }{ \color{blue}{455} })& = & (\frac{-364}{ \color{blue}{455} }x-\frac{2275}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+140& = & -364x-2275 \\\Leftrightarrow & 65x \color{red}{+140} \color{blue}{-140} \color{blue}{+364x} & = & \color{red}{-364x} -2275 \color{blue}{+364x} \color{blue}{-140} \\\Leftrightarrow & 429x& = & -2415 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-2415}{429} \\\Leftrightarrow & x = \frac{-805}{143} & & \\ & V = \left\{ \frac{-805}{143} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{14}& = & \frac{3}{4}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 30 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-30& = & 63x-588 \\\Leftrightarrow & 28x \color{red}{-30} \color{blue}{+30} \color{blue}{-63x} & = & \color{red}{63x} -588 \color{blue}{-63x} \color{blue}{+30} \\\Leftrightarrow & -35x& = & -558 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-558}{-35} \\\Leftrightarrow & x = \frac{558}{35} & & \\ & V = \left\{ \frac{558}{35} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{4}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 112x-672 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-112x} & = & \color{red}{112x} -672 \color{blue}{-112x} \color{blue}{+24} \\\Leftrightarrow & -91x& = & -648 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-648}{-91} \\\Leftrightarrow & x = \frac{648}{91} & & \\ & V = \left\{ \frac{648}{91} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 14x+84 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-14x} & = & \color{red}{14x} +84 \color{blue}{-14x} \color{blue}{+12} \\\Leftrightarrow & 7x& = & 96 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{96}{7} \\\Leftrightarrow & x = \frac{96}{7} & & \\ & V = \left\{ \frac{96}{7} \right\} & \\\end{align}\)
11. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-32}{ \color{blue}{12} }x+\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & -32x+84 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{+32x} & = & \color{red}{-32x} +84 \color{blue}{+32x} \color{blue}{+10} \\\Leftrightarrow & 35x& = & 94 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{94}{35} \\\Leftrightarrow & x = \frac{94}{35} & & \\ & V = \left\{ \frac{94}{35} \right\} & \\\end{align}\)
12. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-32}{ \color{blue}{12} }x+\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+5& = & -32x+48 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{+32x} & = & \color{red}{-32x} +48 \color{blue}{+32x} \color{blue}{-5} \\\Leftrightarrow & 35x& = & 43 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{43}{35} \\\Leftrightarrow & x = \frac{43}{35} & & \\ & V = \left\{ \frac{43}{35} \right\} & \\\end{align}\)