Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{2}{5}x+5 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{44}{ \color{blue}{110} }x+\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+20& = & 44x+550 \\\Leftrightarrow & 55x \color{red}{+20} \color{blue}{-20} \color{blue}{-44x} & = & \color{red}{44x} +550 \color{blue}{-44x} \color{blue}{-20} \\\Leftrightarrow & 11x& = & 530 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{530}{11} \\\Leftrightarrow & x = \frac{530}{11} & & \\ & V = \left\{ \frac{530}{11} \right\} & \\\end{align}\)
2. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 10 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x-\frac{245}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-10& = & -28x-245 \\\Leftrightarrow & 5x \color{red}{-10} \color{blue}{+10} \color{blue}{+28x} & = & \color{red}{-28x} -245 \color{blue}{+28x} \color{blue}{+10} \\\Leftrightarrow & 33x& = & -235 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-235}{33} \\\Leftrightarrow & x = \frac{-235}{33} & & \\ & V = \left\{ \frac{-235}{33} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & -56x-168 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{+56x} & = & \color{red}{-56x} -168 \color{blue}{+56x} \color{blue}{+36} \\\Leftrightarrow & 77x& = & -132 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-132}{77} \\\Leftrightarrow & x = \frac{-12}{7} & & \\ & V = \left\{ \frac{-12}{7} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 33x+132 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-33x} & = & \color{red}{33x} +132 \color{blue}{-33x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & 108 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{108}{-11} \\\Leftrightarrow & x = \frac{-108}{11} & & \\ & V = \left\{ \frac{-108}{11} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{5}{4}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & 105x-252 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{-105x} & = & \color{red}{105x} -252 \color{blue}{-105x} \color{blue}{+24} \\\Leftrightarrow & -77x& = & -228 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-228}{-77} \\\Leftrightarrow & x = \frac{228}{77} & & \\ & V = \left\{ \frac{228}{77} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-80}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & -80x+60 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+80x} & = & \color{red}{-80x} +60 \color{blue}{+80x} \color{blue}{-8} \\\Leftrightarrow & 95x& = & 52 \\\Leftrightarrow & \frac{95x}{ \color{red}{95} }& = & \frac{52}{95} \\\Leftrightarrow & x = \frac{52}{95} & & \\ & V = \left\{ \frac{52}{95} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+9& = & 35x+84 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{-35x} & = & \color{red}{35x} +84 \color{blue}{-35x} \color{blue}{-9} \\\Leftrightarrow & -29x& = & 75 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{75}{-29} \\\Leftrightarrow & x = \frac{-75}{29} & & \\ & V = \left\{ \frac{-75}{29} \right\} & \\\end{align}\)
8. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{4}{5}x-1 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{192}{ \color{blue}{240} }x-\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 192x-240 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-192x} & = & \color{red}{192x} -240 \color{blue}{-192x} \color{blue}{+45} \\\Leftrightarrow & -152x& = & -195 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{-195}{-152} \\\Leftrightarrow & x = \frac{195}{152} & & \\ & V = \left\{ \frac{195}{152} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{3}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 18x+120 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-18x} & = & \color{red}{18x} +120 \color{blue}{-18x} \color{blue}{+4} \\\Leftrightarrow & -13x& = & 124 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{124}{-13} \\\Leftrightarrow & x = \frac{-124}{13} & & \\ & V = \left\{ \frac{-124}{13} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x+\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & 40x+240 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-40x} & = & \color{red}{40x} +240 \color{blue}{-40x} \color{blue}{+15} \\\Leftrightarrow & -24x& = & 255 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{255}{-24} \\\Leftrightarrow & x = \frac{-85}{8} & & \\ & V = \left\{ \frac{-85}{8} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & 24x+288 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-24x} & = & \color{red}{24x} +288 \color{blue}{-24x} \color{blue}{+15} \\\Leftrightarrow & -8x& = & 303 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{303}{-8} \\\Leftrightarrow & x = \frac{-303}{8} & & \\ & V = \left\{ \frac{-303}{8} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 196x-84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-196x} & = & \color{red}{196x} -84 \color{blue}{-196x} \color{blue}{+24} \\\Leftrightarrow & -175x& = & -60 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-60}{-175} \\\Leftrightarrow & x = \frac{12}{35} & & \\ & V = \left\{ \frac{12}{35} \right\} & \\\end{align}\)