Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{9}{ \color{blue}{36} }x+\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-16& = & 9x+180 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{-9x} & = & \color{red}{9x} +180 \color{blue}{-9x} \color{blue}{+16} \\\Leftrightarrow & 3x& = & 196 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{196}{3} \\\Leftrightarrow & x = \frac{196}{3} & & \\ & V = \left\{ \frac{196}{3} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{11}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 36 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-36& = & 44x-660 \\\Leftrightarrow & 33x \color{red}{-36} \color{blue}{+36} \color{blue}{-44x} & = & \color{red}{44x} -660 \color{blue}{-44x} \color{blue}{+36} \\\Leftrightarrow & -11x& = & -624 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-624}{-11} \\\Leftrightarrow & x = \frac{624}{11} & & \\ & V = \left\{ \frac{624}{11} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & -110x+132 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{+110x} & = & \color{red}{-110x} +132 \color{blue}{+110x} \color{blue}{+30} \\\Leftrightarrow & 143x& = & 162 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{162}{143} \\\Leftrightarrow & x = \frac{162}{143} & & \\ & V = \left\{ \frac{162}{143} \right\} & \\\end{align}\)
4. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-104}{ \color{blue}{260} }x+\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-80& = & -104x+520 \\\Leftrightarrow & 65x \color{red}{-80} \color{blue}{+80} \color{blue}{+104x} & = & \color{red}{-104x} +520 \color{blue}{+104x} \color{blue}{+80} \\\Leftrightarrow & 169x& = & 600 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{600}{169} \\\Leftrightarrow & x = \frac{600}{169} & & \\ & V = \left\{ \frac{600}{169} \right\} & \\\end{align}\)
5. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 140 }{ \color{blue}{455} })& = & (\frac{-364}{ \color{blue}{455} }x-\frac{910}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+140& = & -364x-910 \\\Leftrightarrow & 65x \color{red}{+140} \color{blue}{-140} \color{blue}{+364x} & = & \color{red}{-364x} -910 \color{blue}{+364x} \color{blue}{-140} \\\Leftrightarrow & 429x& = & -1050 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-1050}{429} \\\Leftrightarrow & x = \frac{-350}{143} & & \\ & V = \left\{ \frac{-350}{143} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -28x-42 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+28x} & = & \color{red}{-28x} -42 \color{blue}{+28x} \color{blue}{+18} \\\Leftrightarrow & 49x& = & -24 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-24}{49} \\\Leftrightarrow & x = \frac{-24}{49} & & \\ & V = \left\{ \frac{-24}{49} \right\} & \\\end{align}\)
7. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-130}{ \color{blue}{195} }x-\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+45& = & -130x-585 \\\Leftrightarrow & 39x \color{red}{+45} \color{blue}{-45} \color{blue}{+130x} & = & \color{red}{-130x} -585 \color{blue}{+130x} \color{blue}{-45} \\\Leftrightarrow & 169x& = & -630 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-630}{169} \\\Leftrightarrow & x = \frac{-630}{169} & & \\ & V = \left\{ \frac{-630}{169} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{4}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & 21x-168 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & 7x& = & -144 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-144}{7} \\\Leftrightarrow & x = \frac{-144}{7} & & \\ & V = \left\{ \frac{-144}{7} \right\} & \\\end{align}\)
9. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 40x-80 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-40x} & = & \color{red}{40x} -80 \color{blue}{-40x} \color{blue}{-25} \\\Leftrightarrow & -24x& = & -105 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-105}{-24} \\\Leftrightarrow & x = \frac{35}{8} & & \\ & V = \left\{ \frac{35}{8} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 33x-66 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-33x} & = & \color{red}{33x} -66 \color{blue}{-33x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & -90 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-90}{-11} \\\Leftrightarrow & x = \frac{90}{11} & & \\ & V = \left\{ \frac{90}{11} \right\} & \\\end{align}\)
11. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 6 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x-\frac{196}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+6& = & -21x-196 \\\Leftrightarrow & 4x \color{red}{+6} \color{blue}{-6} \color{blue}{+21x} & = & \color{red}{-21x} -196 \color{blue}{+21x} \color{blue}{-6} \\\Leftrightarrow & 25x& = & -202 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{-202}{25} \\\Leftrightarrow & x = \frac{-202}{25} & & \\ & V = \left\{ \frac{-202}{25} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-50& = & -72x+180 \\\Leftrightarrow & 45x \color{red}{-50} \color{blue}{+50} \color{blue}{+72x} & = & \color{red}{-72x} +180 \color{blue}{+72x} \color{blue}{+50} \\\Leftrightarrow & 117x& = & 230 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{230}{117} \\\Leftrightarrow & x = \frac{230}{117} & & \\ & V = \left\{ \frac{230}{117} \right\} & \\\end{align}\)