Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{5}{16}=\frac{5}{6}x-2\)
2. \(\frac{x}{6}+\frac{5}{6}=\frac{-4}{5}x-6\)
3. \(\frac{x}{4}-\frac{3}{14}=\frac{-5}{3}x-2\)
4. \(\frac{x}{5}+\frac{5}{7}=\frac{5}{6}x-7\)
5. \(\frac{x}{7}+\frac{2}{11}=\frac{7}{2}x+5\)
6. \(\frac{x}{3}-\frac{3}{7}=\frac{5}{4}x+8\)
7. \(\frac{x}{4}+\frac{2}{13}=\frac{1}{5}x+8\)
8. \(\frac{x}{2}+\frac{3}{11}=\frac{4}{3}x-7\)
9. \(\frac{x}{2}+\frac{5}{6}=\frac{7}{3}x+8\)
10. \(\frac{x}{7}+\frac{4}{9}=\frac{7}{6}x+2\)
11. \(\frac{x}{4}+\frac{2}{9}=\frac{-2}{5}x-6\)
12. \(\frac{x}{7}-\frac{3}{7}=\frac{1}{2}x-6\)

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

Verbetersleutel

1. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{5}{6}x-2 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 105 }{ \color{blue}{336} })& = & (\frac{280}{ \color{blue}{336} }x-\frac{672}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+105& = & 280x-672 \\\Leftrightarrow & 48x \color{red}{+105} \color{blue}{-105} \color{blue}{-280x} & = & \color{red}{280x} -672 \color{blue}{-280x} \color{blue}{-105} \\\Leftrightarrow & -232x& = & -777 \\\Leftrightarrow & \frac{-232x}{ \color{red}{-232} }& = & \frac{-777}{-232} \\\Leftrightarrow & x = \frac{777}{232} & & \\ & V = \left\{ \frac{777}{232} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -24x-180 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} -180 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 29x& = & -205 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-205}{29} \\\Leftrightarrow & x = \frac{-205}{29} & & \\ & V = \left\{ \frac{-205}{29} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & -140x-168 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+140x} & = & \color{red}{-140x} -168 \color{blue}{+140x} \color{blue}{+18} \\\Leftrightarrow & 161x& = & -150 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-150}{161} \\\Leftrightarrow & x = \frac{-150}{161} & & \\ & V = \left\{ \frac{-150}{161} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{7}& = & \frac{5}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+150& = & 175x-1470 \\\Leftrightarrow & 42x \color{red}{+150} \color{blue}{-150} \color{blue}{-175x} & = & \color{red}{175x} -1470 \color{blue}{-175x} \color{blue}{-150} \\\Leftrightarrow & -133x& = & -1620 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-1620}{-133} \\\Leftrightarrow & x = \frac{1620}{133} & & \\ & V = \left\{ \frac{1620}{133} \right\} & \\\end{align}\)
5. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{7}{2}x+5 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{539}{ \color{blue}{154} }x+\frac{770}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 539x+770 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-539x} & = & \color{red}{539x} +770 \color{blue}{-539x} \color{blue}{-28} \\\Leftrightarrow & -517x& = & 742 \\\Leftrightarrow & \frac{-517x}{ \color{red}{-517} }& = & \frac{742}{-517} \\\Leftrightarrow & x = \frac{-742}{517} & & \\ & V = \left\{ \frac{-742}{517} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{5}{4}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 105x+672 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-105x} & = & \color{red}{105x} +672 \color{blue}{-105x} \color{blue}{+36} \\\Leftrightarrow & -77x& = & 708 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{708}{-77} \\\Leftrightarrow & x = \frac{-708}{77} & & \\ & V = \left\{ \frac{-708}{77} \right\} & \\\end{align}\)
7. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{52}{ \color{blue}{260} }x+\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+40& = & 52x+2080 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{-52x} & = & \color{red}{52x} +2080 \color{blue}{-52x} \color{blue}{-40} \\\Leftrightarrow & 13x& = & 2040 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{2040}{13} \\\Leftrightarrow & x = \frac{2040}{13} & & \\ & V = \left\{ \frac{2040}{13} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{11}& = & \frac{4}{3}x-7 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 18 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x-\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+18& = & 88x-462 \\\Leftrightarrow & 33x \color{red}{+18} \color{blue}{-18} \color{blue}{-88x} & = & \color{red}{88x} -462 \color{blue}{-88x} \color{blue}{-18} \\\Leftrightarrow & -55x& = & -480 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-480}{-55} \\\Leftrightarrow & x = \frac{96}{11} & & \\ & V = \left\{ \frac{96}{11} \right\} & \\\end{align}\)
9. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{14}{ \color{blue}{6} }x+\frac{48}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 14x+48 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-14x} & = & \color{red}{14x} +48 \color{blue}{-14x} \color{blue}{-5} \\\Leftrightarrow & -11x& = & 43 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{43}{-11} \\\Leftrightarrow & x = \frac{-43}{11} & & \\ & V = \left\{ \frac{-43}{11} \right\} & \\\end{align}\)
10. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{7}{6}x+2 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{147}{ \color{blue}{126} }x+\frac{252}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 147x+252 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-147x} & = & \color{red}{147x} +252 \color{blue}{-147x} \color{blue}{-56} \\\Leftrightarrow & -129x& = & 196 \\\Leftrightarrow & \frac{-129x}{ \color{red}{-129} }& = & \frac{196}{-129} \\\Leftrightarrow & x = \frac{-196}{129} & & \\ & V = \left\{ \frac{-196}{129} \right\} & \\\end{align}\)
11. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-72}{ \color{blue}{180} }x-\frac{1080}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+40& = & -72x-1080 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{+72x} & = & \color{red}{-72x} -1080 \color{blue}{+72x} \color{blue}{-40} \\\Leftrightarrow & 117x& = & -1120 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-1120}{117} \\\Leftrightarrow & x = \frac{-1120}{117} & & \\ & V = \left\{ \frac{-1120}{117} \right\} & \\\end{align}\)
12. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x-84 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} -84 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & -78 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-78}{-5} \\\Leftrightarrow & x = \frac{78}{5} & & \\ & V = \left\{ \frac{78}{5} \right\} & \\\end{align}\)