Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & -20x+150 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{+20x} & = & \color{red}{-20x} +150 \color{blue}{+20x} \color{blue}{-4} \\\Leftrightarrow & 35x& = & 146 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{146}{35} \\\Leftrightarrow & x = \frac{146}{35} & & \\ & V = \left\{ \frac{146}{35} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 3 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-3& = & 7x+28 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-7x} & = & \color{red}{7x} +28 \color{blue}{-7x} \color{blue}{+3} \\\Leftrightarrow & -5x& = & 31 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{31}{-5} \\\Leftrightarrow & x = \frac{-31}{5} & & \\ & V = \left\{ \frac{-31}{5} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{9}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 20 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x-\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+20& = & 12x-216 \\\Leftrightarrow & 9x \color{red}{+20} \color{blue}{-20} \color{blue}{-12x} & = & \color{red}{12x} -216 \color{blue}{-12x} \color{blue}{-20} \\\Leftrightarrow & -3x& = & -236 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-236}{-3} \\\Leftrightarrow & x = \frac{236}{3} & & \\ & V = \left\{ \frac{236}{3} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x-\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & -264x-2310 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{+264x} & = & \color{red}{-264x} -2310 \color{blue}{+264x} \color{blue}{+120} \\\Leftrightarrow & 319x& = & -2190 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-2190}{319} \\\Leftrightarrow & x = \frac{-2190}{319} & & \\ & V = \left\{ \frac{-2190}{319} \right\} & \\\end{align}\)
5. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{-7}{4}x+5 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 40 }{ \color{blue}{260} })& = & (\frac{-455}{ \color{blue}{260} }x+\frac{1300}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-40& = & -455x+1300 \\\Leftrightarrow & 52x \color{red}{-40} \color{blue}{+40} \color{blue}{+455x} & = & \color{red}{-455x} +1300 \color{blue}{+455x} \color{blue}{+40} \\\Leftrightarrow & 507x& = & 1340 \\\Leftrightarrow & \frac{507x}{ \color{red}{507} }& = & \frac{1340}{507} \\\Leftrightarrow & x = \frac{1340}{507} & & \\ & V = \left\{ \frac{1340}{507} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{14}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 15 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-15& = & -28x+126 \\\Leftrightarrow & 21x \color{red}{-15} \color{blue}{+15} \color{blue}{+28x} & = & \color{red}{-28x} +126 \color{blue}{+28x} \color{blue}{+15} \\\Leftrightarrow & 49x& = & 141 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{141}{49} \\\Leftrightarrow & x = \frac{141}{49} & & \\ & V = \left\{ \frac{141}{49} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 147x-294 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-147x} & = & \color{red}{147x} -294 \color{blue}{-147x} \color{blue}{-18} \\\Leftrightarrow & -133x& = & -312 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-312}{-133} \\\Leftrightarrow & x = \frac{312}{133} & & \\ & V = \left\{ \frac{312}{133} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{2}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{28}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & 28x+168 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{-28x} & = & \color{red}{28x} +168 \color{blue}{-28x} \color{blue}{+9} \\\Leftrightarrow & -7x& = & 177 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{177}{-7} \\\Leftrightarrow & x = \frac{-177}{7} & & \\ & V = \left\{ \frac{-177}{7} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{13}& = & \frac{5}{6}x-1 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{325}{ \color{blue}{390} }x-\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+150& = & 325x-390 \\\Leftrightarrow & 78x \color{red}{+150} \color{blue}{-150} \color{blue}{-325x} & = & \color{red}{325x} -390 \color{blue}{-325x} \color{blue}{-150} \\\Leftrightarrow & -247x& = & -540 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-540}{-247} \\\Leftrightarrow & x = \frac{540}{247} & & \\ & V = \left\{ \frac{540}{247} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-30& = & 35x+490 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-35x} & = & \color{red}{35x} +490 \color{blue}{-35x} \color{blue}{+30} \\\Leftrightarrow & -21x& = & 520 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{520}{-21} \\\Leftrightarrow & x = \frac{-520}{21} & & \\ & V = \left\{ \frac{-520}{21} \right\} & \\\end{align}\)
11. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{13}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 100 }{ \color{blue}{260} })& = & (\frac{312}{ \color{blue}{260} }x+\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+100& = & 312x+1560 \\\Leftrightarrow & 65x \color{red}{+100} \color{blue}{-100} \color{blue}{-312x} & = & \color{red}{312x} +1560 \color{blue}{-312x} \color{blue}{-100} \\\Leftrightarrow & -247x& = & 1460 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{1460}{-247} \\\Leftrightarrow & x = \frac{-1460}{247} & & \\ & V = \left\{ \frac{-1460}{247} \right\} & \\\end{align}\)
12. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 80 }{ \color{blue}{180} })& = & (\frac{-252}{ \color{blue}{180} }x-\frac{1080}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-80& = & -252x-1080 \\\Leftrightarrow & 45x \color{red}{-80} \color{blue}{+80} \color{blue}{+252x} & = & \color{red}{-252x} -1080 \color{blue}{+252x} \color{blue}{+80} \\\Leftrightarrow & 297x& = & -1000 \\\Leftrightarrow & \frac{297x}{ \color{red}{297} }& = & \frac{-1000}{297} \\\Leftrightarrow & x = \frac{-1000}{297} & & \\ & V = \left\{ \frac{-1000}{297} \right\} & \\\end{align}\)