Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{2}{13}=\frac{-5}{3}x-1\)
2. \(\frac{x}{3}-\frac{3}{10}=\frac{1}{2}x-2\)
3. \(\frac{x}{2}-\frac{3}{14}=\frac{-2}{3}x-5\)
4. \(\frac{x}{3}-\frac{3}{8}=\frac{-7}{4}x+8\)
5. \(\frac{x}{3}-\frac{5}{16}=\frac{3}{4}x+6\)
6. \(\frac{x}{5}+\frac{2}{15}=\frac{7}{4}x-1\)
7. \(\frac{x}{5}-\frac{3}{7}=\frac{-7}{4}x-3\)
8. \(\frac{x}{4}+\frac{2}{9}=\frac{-8}{3}x-4\)
9. \(\frac{x}{4}-\frac{3}{13}=\frac{4}{3}x-3\)
10. \(\frac{x}{7}-\frac{5}{6}=\frac{-5}{3}x+3\)
11. \(\frac{x}{7}+\frac{2}{11}=\frac{1}{6}x+2\)
12. \(\frac{x}{7}-\frac{4}{11}=\frac{-7}{4}x+1\)

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

Verbetersleutel

1. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & -130x-78 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{+130x} & = & \color{red}{-130x} -78 \color{blue}{+130x} \color{blue}{-12} \\\Leftrightarrow & 169x& = & -90 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-90}{169} \\\Leftrightarrow & x = \frac{-90}{169} & & \\ & V = \left\{ \frac{-90}{169} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 15x-60 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-15x} & = & \color{red}{15x} -60 \color{blue}{-15x} \color{blue}{+9} \\\Leftrightarrow & -5x& = & -51 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-51}{-5} \\\Leftrightarrow & x = \frac{51}{5} & & \\ & V = \left\{ \frac{51}{5} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & -28x-210 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{+28x} & = & \color{red}{-28x} -210 \color{blue}{+28x} \color{blue}{+9} \\\Leftrightarrow & 49x& = & -201 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-201}{49} \\\Leftrightarrow & x = \frac{-201}{49} & & \\ & V = \left\{ \frac{-201}{49} \right\} & \\\end{align}\)
4. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{-7}{4}x+8 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-42}{ \color{blue}{24} }x+\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & -42x+192 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{+42x} & = & \color{red}{-42x} +192 \color{blue}{+42x} \color{blue}{+9} \\\Leftrightarrow & 50x& = & 201 \\\Leftrightarrow & \frac{50x}{ \color{red}{50} }& = & \frac{201}{50} \\\Leftrightarrow & x = \frac{201}{50} & & \\ & V = \left\{ \frac{201}{50} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{3}{4}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{36}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & 36x+288 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-36x} & = & \color{red}{36x} +288 \color{blue}{-36x} \color{blue}{+15} \\\Leftrightarrow & -20x& = & 303 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{303}{-20} \\\Leftrightarrow & x = \frac{-303}{20} & & \\ & V = \left\{ \frac{-303}{20} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{7}{4}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{105}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+8& = & 105x-60 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{-105x} & = & \color{red}{105x} -60 \color{blue}{-105x} \color{blue}{-8} \\\Leftrightarrow & -93x& = & -68 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{-68}{-93} \\\Leftrightarrow & x = \frac{68}{93} & & \\ & V = \left\{ \frac{68}{93} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-60& = & -245x-420 \\\Leftrightarrow & 28x \color{red}{-60} \color{blue}{+60} \color{blue}{+245x} & = & \color{red}{-245x} -420 \color{blue}{+245x} \color{blue}{+60} \\\Leftrightarrow & 273x& = & -360 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{-360}{273} \\\Leftrightarrow & x = \frac{-120}{91} & & \\ & V = \left\{ \frac{-120}{91} \right\} & \\\end{align}\)
8. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-8}{3}x-4 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-96}{ \color{blue}{36} }x-\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & -96x-144 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+96x} & = & \color{red}{-96x} -144 \color{blue}{+96x} \color{blue}{-8} \\\Leftrightarrow & 105x& = & -152 \\\Leftrightarrow & \frac{105x}{ \color{red}{105} }& = & \frac{-152}{105} \\\Leftrightarrow & x = \frac{-152}{105} & & \\ & V = \left\{ \frac{-152}{105} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x-\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & 208x-468 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{-208x} & = & \color{red}{208x} -468 \color{blue}{-208x} \color{blue}{+36} \\\Leftrightarrow & -169x& = & -432 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-432}{-169} \\\Leftrightarrow & x = \frac{432}{169} & & \\ & V = \left\{ \frac{432}{169} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & -70x+126 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{+70x} & = & \color{red}{-70x} +126 \color{blue}{+70x} \color{blue}{+35} \\\Leftrightarrow & 76x& = & 161 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{161}{76} \\\Leftrightarrow & x = \frac{161}{76} & & \\ & V = \left\{ \frac{161}{76} \right\} & \\\end{align}\)
11. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{1}{6}x+2 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 84 }{ \color{blue}{462} })& = & (\frac{77}{ \color{blue}{462} }x+\frac{924}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+84& = & 77x+924 \\\Leftrightarrow & 66x \color{red}{+84} \color{blue}{-84} \color{blue}{-77x} & = & \color{red}{77x} +924 \color{blue}{-77x} \color{blue}{-84} \\\Leftrightarrow & -11x& = & 840 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{840}{-11} \\\Leftrightarrow & x = \frac{-840}{11} & & \\ & V = \left\{ \frac{-840}{11} \right\} & \\\end{align}\)
12. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{-7}{4}x+1 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }- \frac{ 112 }{ \color{blue}{308} })& = & (\frac{-539}{ \color{blue}{308} }x+\frac{308}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x-112& = & -539x+308 \\\Leftrightarrow & 44x \color{red}{-112} \color{blue}{+112} \color{blue}{+539x} & = & \color{red}{-539x} +308 \color{blue}{+539x} \color{blue}{+112} \\\Leftrightarrow & 583x& = & 420 \\\Leftrightarrow & \frac{583x}{ \color{red}{583} }& = & \frac{420}{583} \\\Leftrightarrow & x = \frac{420}{583} & & \\ & V = \left\{ \frac{420}{583} \right\} & \\\end{align}\)