Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{4}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{264}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & 264x+2640 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{-264x} & = & \color{red}{264x} +2640 \color{blue}{-264x} \color{blue}{+120} \\\Leftrightarrow & -209x& = & 2760 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{2760}{-209} \\\Leftrightarrow & x = \frac{-2760}{209} & & \\ & V = \left\{ \frac{-2760}{209} \right\} & \\\end{align}\)
2. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 22x+264 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-22x} & = & \color{red}{22x} +264 \color{blue}{-22x} \color{blue}{+30} \\\Leftrightarrow & 11x& = & 294 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{294}{11} \\\Leftrightarrow & x = \frac{294}{11} & & \\ & V = \left\{ \frac{294}{11} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -56x-672 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+56x} & = & \color{red}{-56x} -672 \color{blue}{+56x} \color{blue}{-48} \\\Leftrightarrow & 77x& = & -720 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-720}{77} \\\Leftrightarrow & x = \frac{-720}{77} & & \\ & V = \left\{ \frac{-720}{77} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & -44x-132 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+44x} & = & \color{red}{-44x} -132 \color{blue}{+44x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & -108 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-108}{77} \\\Leftrightarrow & x = \frac{-108}{77} & & \\ & V = \left\{ \frac{-108}{77} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x+\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & 52x+1248 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{-52x} & = & \color{red}{52x} +1248 \color{blue}{-52x} \color{blue}{+60} \\\Leftrightarrow & -13x& = & 1308 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{1308}{-13} \\\Leftrightarrow & x = \frac{-1308}{13} & & \\ & V = \left\{ \frac{-1308}{13} \right\} & \\\end{align}\)
6. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{7}{5}x+5 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{28}{ \color{blue}{20} }x+\frac{100}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & 28x+100 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{-28x} & = & \color{red}{28x} +100 \color{blue}{-28x} \color{blue}{-6} \\\Leftrightarrow & -23x& = & 94 \\\Leftrightarrow & \frac{-23x}{ \color{red}{-23} }& = & \frac{94}{-23} \\\Leftrightarrow & x = \frac{-94}{23} & & \\ & V = \left\{ \frac{-94}{23} \right\} & \\\end{align}\)
7. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{5}{4}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{195}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-48& = & 195x+624 \\\Leftrightarrow & 52x \color{red}{-48} \color{blue}{+48} \color{blue}{-195x} & = & \color{red}{195x} +624 \color{blue}{-195x} \color{blue}{+48} \\\Leftrightarrow & -143x& = & 672 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{672}{-143} \\\Leftrightarrow & x = \frac{-672}{143} & & \\ & V = \left\{ \frac{-672}{143} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{-3}{4}x+6 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x+\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+36& = & -117x+936 \\\Leftrightarrow & 52x \color{red}{+36} \color{blue}{-36} \color{blue}{+117x} & = & \color{red}{-117x} +936 \color{blue}{+117x} \color{blue}{-36} \\\Leftrightarrow & 169x& = & 900 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{900}{169} \\\Leftrightarrow & x = \frac{900}{169} & & \\ & V = \left\{ \frac{900}{169} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{84}{ \color{blue}{140} }x+\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 84x+280 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-84x} & = & \color{red}{84x} +280 \color{blue}{-84x} \color{blue}{-40} \\\Leftrightarrow & -49x& = & 240 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{240}{-49} \\\Leftrightarrow & x = \frac{-240}{49} & & \\ & V = \left\{ \frac{-240}{49} \right\} & \\\end{align}\)
10. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{-3}{4}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-60& = & -117x-156 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{+117x} & = & \color{red}{-117x} -156 \color{blue}{+117x} \color{blue}{+60} \\\Leftrightarrow & 169x& = & -96 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-96}{169} \\\Leftrightarrow & x = \frac{-96}{169} & & \\ & V = \left\{ \frac{-96}{169} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & 22x+66 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{-22x} & = & \color{red}{22x} +66 \color{blue}{-22x} \color{blue}{-12} \\\Leftrightarrow & 11x& = & 54 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{54}{11} \\\Leftrightarrow & x = \frac{54}{11} & & \\ & V = \left\{ \frac{54}{11} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{-42}{ \color{blue}{105} }x+\frac{315}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-45& = & -42x+315 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{+42x} & = & \color{red}{-42x} +315 \color{blue}{+42x} \color{blue}{+45} \\\Leftrightarrow & 77x& = & 360 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{360}{77} \\\Leftrightarrow & x = \frac{360}{77} & & \\ & V = \left\{ \frac{360}{77} \right\} & \\\end{align}\)