Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{13}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 60 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+60& = & 52x-156 \\\Leftrightarrow & 39x \color{red}{+60} \color{blue}{-60} \color{blue}{-52x} & = & \color{red}{52x} -156 \color{blue}{-52x} \color{blue}{-60} \\\Leftrightarrow & -13x& = & -216 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-216}{-13} \\\Leftrightarrow & x = \frac{216}{13} & & \\ & V = \left\{ \frac{216}{13} \right\} & \\\end{align}\)
2. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & 16x+400 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{-16x} & = & \color{red}{16x} +400 \color{blue}{-16x} \color{blue}{+25} \\\Leftrightarrow & 24x& = & 425 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{425}{24} \\\Leftrightarrow & x = \frac{425}{24} & & \\ & V = \left\{ \frac{425}{24} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 28x+672 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-28x} & = & \color{red}{28x} +672 \color{blue}{-28x} \color{blue}{+36} \\\Leftrightarrow & -7x& = & 708 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{708}{-7} \\\Leftrightarrow & x = \frac{-708}{7} & & \\ & V = \left\{ \frac{-708}{7} \right\} & \\\end{align}\)
4. \(\text{40 is het kleinste gemene veelvoud van 4, 8 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{8}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{40.} (\frac{10x}{ \color{blue}{40} }- \frac{ 25 }{ \color{blue}{40} })& = & (\frac{48}{ \color{blue}{40} }x+\frac{160}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 10x-25& = & 48x+160 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-48x} & = & \color{red}{48x} +160 \color{blue}{-48x} \color{blue}{+25} \\\Leftrightarrow & -38x& = & 185 \\\Leftrightarrow & \frac{-38x}{ \color{red}{-38} }& = & \frac{185}{-38} \\\Leftrightarrow & x = \frac{-185}{38} & & \\ & V = \left\{ \frac{-185}{38} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-352}{ \color{blue}{132} }x-\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-60& = & -352x-1056 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{+352x} & = & \color{red}{-352x} -1056 \color{blue}{+352x} \color{blue}{+60} \\\Leftrightarrow & 385x& = & -996 \\\Leftrightarrow & \frac{385x}{ \color{red}{385} }& = & \frac{-996}{385} \\\Leftrightarrow & x = \frac{-996}{385} & & \\ & V = \left\{ \frac{-996}{385} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{14}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+9& = & 14x+252 \\\Leftrightarrow & 21x \color{red}{+9} \color{blue}{-9} \color{blue}{-14x} & = & \color{red}{14x} +252 \color{blue}{-14x} \color{blue}{-9} \\\Leftrightarrow & 7x& = & 243 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{243}{7} \\\Leftrightarrow & x = \frac{243}{7} & & \\ & V = \left\{ \frac{243}{7} \right\} & \\\end{align}\)
7. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x+\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+16& = & -27x+288 \\\Leftrightarrow & 12x \color{red}{+16} \color{blue}{-16} \color{blue}{+27x} & = & \color{red}{-27x} +288 \color{blue}{+27x} \color{blue}{-16} \\\Leftrightarrow & 39x& = & 272 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{272}{39} \\\Leftrightarrow & x = \frac{272}{39} & & \\ & V = \left\{ \frac{272}{39} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 5x-240 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-5x} & = & \color{red}{5x} -240 \color{blue}{-5x} \color{blue}{-25} \\\Leftrightarrow & x& = & -265 \\ & V = \left\{ -265 \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & 84x-490 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{-84x} & = & \color{red}{84x} -490 \color{blue}{-84x} \color{blue}{-20} \\\Leftrightarrow & -49x& = & -510 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-510}{-49} \\\Leftrightarrow & x = \frac{510}{49} & & \\ & V = \left\{ \frac{510}{49} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & -264x+1980 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{+264x} & = & \color{red}{-264x} +1980 \color{blue}{+264x} \color{blue}{-120} \\\Leftrightarrow & 319x& = & 1860 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{1860}{319} \\\Leftrightarrow & x = \frac{1860}{319} & & \\ & V = \left\{ \frac{1860}{319} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -42x-120 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+42x} & = & \color{red}{-42x} -120 \color{blue}{+42x} \color{blue}{-25} \\\Leftrightarrow & 47x& = & -145 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-145}{47} \\\Leftrightarrow & x = \frac{-145}{47} & & \\ & V = \left\{ \frac{-145}{47} \right\} & \\\end{align}\)
12. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 63 }{ \color{blue}{336} })& = & (\frac{448}{ \color{blue}{336} }x+\frac{336}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+63& = & 448x+336 \\\Leftrightarrow & 48x \color{red}{+63} \color{blue}{-63} \color{blue}{-448x} & = & \color{red}{448x} +336 \color{blue}{-448x} \color{blue}{-63} \\\Leftrightarrow & -400x& = & 273 \\\Leftrightarrow & \frac{-400x}{ \color{red}{-400} }& = & \frac{273}{-400} \\\Leftrightarrow & x = \frac{-273}{400} & & \\ & V = \left\{ \frac{-273}{400} \right\} & \\\end{align}\)