Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}-\frac{4}{9}=\frac{1}{2}x-8\)
2. \(\frac{x}{4}-\frac{5}{6}=\frac{-5}{3}x-8\)
3. \(\frac{x}{6}-\frac{2}{15}=\frac{-4}{5}x-3\)
4. \(\frac{x}{4}+\frac{3}{14}=\frac{6}{5}x-3\)
5. \(\frac{x}{3}+\frac{3}{14}=\frac{1}{2}x+3\)
6. \(\frac{x}{2}-\frac{4}{11}=\frac{-4}{5}x-1\)
7. \(\frac{x}{2}-\frac{4}{9}=\frac{6}{5}x+8\)
8. \(\frac{x}{2}-\frac{5}{13}=\frac{-4}{5}x-2\)
9. \(\frac{x}{7}-\frac{3}{16}=\frac{1}{4}x-2\)
10. \(\frac{x}{7}+\frac{2}{9}=\frac{-2}{3}x-4\)
11. \(\frac{x}{4}-\frac{2}{7}=\frac{7}{3}x-7\)
12. \(\frac{x}{4}-\frac{4}{15}=\frac{5}{3}x-3\)

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

Verbetersleutel

1. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x-\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-56& = & 63x-1008 \\\Leftrightarrow & 18x \color{red}{-56} \color{blue}{+56} \color{blue}{-63x} & = & \color{red}{63x} -1008 \color{blue}{-63x} \color{blue}{+56} \\\Leftrightarrow & -45x& = & -952 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-952}{-45} \\\Leftrightarrow & x = \frac{952}{45} & & \\ & V = \left\{ \frac{952}{45} \right\} & \\\end{align}\)
2. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-20}{ \color{blue}{12} }x-\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-10& = & -20x-96 \\\Leftrightarrow & 3x \color{red}{-10} \color{blue}{+10} \color{blue}{+20x} & = & \color{red}{-20x} -96 \color{blue}{+20x} \color{blue}{+10} \\\Leftrightarrow & 23x& = & -86 \\\Leftrightarrow & \frac{23x}{ \color{red}{23} }& = & \frac{-86}{23} \\\Leftrightarrow & x = \frac{-86}{23} & & \\ & V = \left\{ \frac{-86}{23} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -24x-90 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+24x} & = & \color{red}{-24x} -90 \color{blue}{+24x} \color{blue}{+4} \\\Leftrightarrow & 29x& = & -86 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-86}{29} \\\Leftrightarrow & x = \frac{-86}{29} & & \\ & V = \left\{ \frac{-86}{29} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 30 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+30& = & 168x-420 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-168x} & = & \color{red}{168x} -420 \color{blue}{-168x} \color{blue}{-30} \\\Leftrightarrow & -133x& = & -450 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-450}{-133} \\\Leftrightarrow & x = \frac{450}{133} & & \\ & V = \left\{ \frac{450}{133} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{14}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+9& = & 21x+126 \\\Leftrightarrow & 14x \color{red}{+9} \color{blue}{-9} \color{blue}{-21x} & = & \color{red}{21x} +126 \color{blue}{-21x} \color{blue}{-9} \\\Leftrightarrow & -7x& = & 117 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{117}{-7} \\\Leftrightarrow & x = \frac{-117}{7} & & \\ & V = \left\{ \frac{-117}{7} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{-88}{ \color{blue}{110} }x-\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-40& = & -88x-110 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{+88x} & = & \color{red}{-88x} -110 \color{blue}{+88x} \color{blue}{+40} \\\Leftrightarrow & 143x& = & -70 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-70}{143} \\\Leftrightarrow & x = \frac{-70}{143} & & \\ & V = \left\{ \frac{-70}{143} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-40& = & 108x+720 \\\Leftrightarrow & 45x \color{red}{-40} \color{blue}{+40} \color{blue}{-108x} & = & \color{red}{108x} +720 \color{blue}{-108x} \color{blue}{+40} \\\Leftrightarrow & -63x& = & 760 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{760}{-63} \\\Leftrightarrow & x = \frac{-760}{63} & & \\ & V = \left\{ \frac{-760}{63} \right\} & \\\end{align}\)
8. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 50 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x-\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-50& = & -104x-260 \\\Leftrightarrow & 65x \color{red}{-50} \color{blue}{+50} \color{blue}{+104x} & = & \color{red}{-104x} -260 \color{blue}{+104x} \color{blue}{+50} \\\Leftrightarrow & 169x& = & -210 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-210}{169} \\\Leftrightarrow & x = \frac{-210}{169} & & \\ & V = \left\{ \frac{-210}{169} \right\} & \\\end{align}\)
9. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{1}{4}x-2 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 21 }{ \color{blue}{112} })& = & (\frac{28}{ \color{blue}{112} }x-\frac{224}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-21& = & 28x-224 \\\Leftrightarrow & 16x \color{red}{-21} \color{blue}{+21} \color{blue}{-28x} & = & \color{red}{28x} -224 \color{blue}{-28x} \color{blue}{+21} \\\Leftrightarrow & -12x& = & -203 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{-203}{-12} \\\Leftrightarrow & x = \frac{203}{12} & & \\ & V = \left\{ \frac{203}{12} \right\} & \\\end{align}\)
10. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }+ \frac{ 14 }{ \color{blue}{63} })& = & (\frac{-42}{ \color{blue}{63} }x-\frac{252}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x+14& = & -42x-252 \\\Leftrightarrow & 9x \color{red}{+14} \color{blue}{-14} \color{blue}{+42x} & = & \color{red}{-42x} -252 \color{blue}{+42x} \color{blue}{-14} \\\Leftrightarrow & 51x& = & -266 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-266}{51} \\\Leftrightarrow & x = \frac{-266}{51} & & \\ & V = \left\{ \frac{-266}{51} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 196x-588 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-196x} & = & \color{red}{196x} -588 \color{blue}{-196x} \color{blue}{+24} \\\Leftrightarrow & -175x& = & -564 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-564}{-175} \\\Leftrightarrow & x = \frac{564}{175} & & \\ & V = \left\{ \frac{564}{175} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{5}{3}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{100}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 100x-180 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-100x} & = & \color{red}{100x} -180 \color{blue}{-100x} \color{blue}{+16} \\\Leftrightarrow & -85x& = & -164 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{-164}{-85} \\\Leftrightarrow & x = \frac{164}{85} & & \\ & V = \left\{ \frac{164}{85} \right\} & \\\end{align}\)