Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 42x+1050 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-42x} & = & \color{red}{42x} +1050 \color{blue}{-42x} \color{blue}{-120} \\\Leftrightarrow & -7x& = & 930 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{930}{-7} \\\Leftrightarrow & x = \frac{-930}{7} & & \\ & V = \left\{ \frac{-930}{7} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -126x-630 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+126x} & = & \color{red}{-126x} -630 \color{blue}{+126x} \color{blue}{+40} \\\Leftrightarrow & 141x& = & -590 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-590}{141} \\\Leftrightarrow & x = \frac{-590}{141} & & \\ & V = \left\{ \frac{-590}{141} \right\} & \\\end{align}\)
3. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-144}{ \color{blue}{180} }x+\frac{180}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+40& = & -144x+180 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{+144x} & = & \color{red}{-144x} +180 \color{blue}{+144x} \color{blue}{-40} \\\Leftrightarrow & 189x& = & 140 \\\Leftrightarrow & \frac{189x}{ \color{red}{189} }& = & \frac{140}{189} \\\Leftrightarrow & x = \frac{20}{27} & & \\ & V = \left\{ \frac{20}{27} \right\} & \\\end{align}\)
4. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{1}{4}x-8 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 100 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x-\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-100& = & 65x-2080 \\\Leftrightarrow & 52x \color{red}{-100} \color{blue}{+100} \color{blue}{-65x} & = & \color{red}{65x} -2080 \color{blue}{-65x} \color{blue}{+100} \\\Leftrightarrow & -13x& = & -1980 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-1980}{-13} \\\Leftrightarrow & x = \frac{1980}{13} & & \\ & V = \left\{ \frac{1980}{13} \right\} & \\\end{align}\)
5. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{8}& = & \frac{5}{4}x-6 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 15 }{ \color{blue}{40} })& = & (\frac{50}{ \color{blue}{40} }x-\frac{240}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+15& = & 50x-240 \\\Leftrightarrow & 8x \color{red}{+15} \color{blue}{-15} \color{blue}{-50x} & = & \color{red}{50x} -240 \color{blue}{-50x} \color{blue}{-15} \\\Leftrightarrow & -42x& = & -255 \\\Leftrightarrow & \frac{-42x}{ \color{red}{-42} }& = & \frac{-255}{-42} \\\Leftrightarrow & x = \frac{85}{14} & & \\ & V = \left\{ \frac{85}{14} \right\} & \\\end{align}\)
6. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 10 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x+\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+10& = & -21x+84 \\\Leftrightarrow & 4x \color{red}{+10} \color{blue}{-10} \color{blue}{+21x} & = & \color{red}{-21x} +84 \color{blue}{+21x} \color{blue}{-10} \\\Leftrightarrow & 25x& = & 74 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{74}{25} \\\Leftrightarrow & x = \frac{74}{25} & & \\ & V = \left\{ \frac{74}{25} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{-8}{3}x-5 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-48}{ \color{blue}{18} }x-\frac{90}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & -48x-90 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{+48x} & = & \color{red}{-48x} -90 \color{blue}{+48x} \color{blue}{+4} \\\Leftrightarrow & 57x& = & -86 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-86}{57} \\\Leftrightarrow & x = \frac{-86}{57} & & \\ & V = \left\{ \frac{-86}{57} \right\} & \\\end{align}\)
8. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{5}{3}x-4 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{80}{ \color{blue}{48} }x-\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 80x-192 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-80x} & = & \color{red}{80x} -192 \color{blue}{-80x} \color{blue}{-15} \\\Leftrightarrow & -56x& = & -207 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{-207}{-56} \\\Leftrightarrow & x = \frac{207}{56} & & \\ & V = \left\{ \frac{207}{56} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-20& = & 65x-780 \\\Leftrightarrow & 26x \color{red}{-20} \color{blue}{+20} \color{blue}{-65x} & = & \color{red}{65x} -780 \color{blue}{-65x} \color{blue}{+20} \\\Leftrightarrow & -39x& = & -760 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-760}{-39} \\\Leftrightarrow & x = \frac{760}{39} & & \\ & V = \left\{ \frac{760}{39} \right\} & \\\end{align}\)
10. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{5}{2}x-1 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 70 }{ \color{blue}{154} })& = & (\frac{385}{ \color{blue}{154} }x-\frac{154}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+70& = & 385x-154 \\\Leftrightarrow & 22x \color{red}{+70} \color{blue}{-70} \color{blue}{-385x} & = & \color{red}{385x} -154 \color{blue}{-385x} \color{blue}{-70} \\\Leftrightarrow & -363x& = & -224 \\\Leftrightarrow & \frac{-363x}{ \color{red}{-363} }& = & \frac{-224}{-363} \\\Leftrightarrow & x = \frac{224}{363} & & \\ & V = \left\{ \frac{224}{363} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & -24x+180 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+24x} & = & \color{red}{-24x} +180 \color{blue}{+24x} \color{blue}{+25} \\\Leftrightarrow & 29x& = & 205 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{205}{29} \\\Leftrightarrow & x = \frac{205}{29} & & \\ & V = \left\{ \frac{205}{29} \right\} & \\\end{align}\)
12. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{-4}{ \color{blue}{6} }x+\frac{6}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & -4x+6 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{+4x} & = & \color{red}{-4x} +6 \color{blue}{+4x} \color{blue}{-5} \\\Leftrightarrow & 7x& = & 1 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{1}{7} \\\Leftrightarrow & x = \frac{1}{7} & & \\ & V = \left\{ \frac{1}{7} \right\} & \\\end{align}\)