Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{5}{6}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{325}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-120& = & 325x-1170 \\\Leftrightarrow & 78x \color{red}{-120} \color{blue}{+120} \color{blue}{-325x} & = & \color{red}{325x} -1170 \color{blue}{-325x} \color{blue}{+120} \\\Leftrightarrow & -247x& = & -1050 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-1050}{-247} \\\Leftrightarrow & x = \frac{1050}{247} & & \\ & V = \left\{ \frac{1050}{247} \right\} & \\\end{align}\)
2. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }+ \frac{ 20 }{ \color{blue}{45} })& = & (\frac{-36}{ \color{blue}{45} }x+\frac{270}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x+20& = & -36x+270 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+36x} & = & \color{red}{-36x} +270 \color{blue}{+36x} \color{blue}{-20} \\\Leftrightarrow & 51x& = & 250 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{250}{51} \\\Leftrightarrow & x = \frac{250}{51} & & \\ & V = \left\{ \frac{250}{51} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 15x-90 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} -90 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & -86 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-86}{-5} \\\Leftrightarrow & x = \frac{86}{5} & & \\ & V = \left\{ \frac{86}{5} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 15x-210 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-15x} & = & \color{red}{15x} -210 \color{blue}{-15x} \color{blue}{-9} \\\Leftrightarrow & -5x& = & -219 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-219}{-5} \\\Leftrightarrow & x = \frac{219}{5} & & \\ & V = \left\{ \frac{219}{5} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{5}{6}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & 25x+180 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-25x} & = & \color{red}{25x} +180 \color{blue}{-25x} \color{blue}{+9} \\\Leftrightarrow & -19x& = & 189 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{189}{-19} \\\Leftrightarrow & x = \frac{-189}{19} & & \\ & V = \left\{ \frac{-189}{19} \right\} & \\\end{align}\)
6. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x-\frac{30}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x-5& = & 3x-30 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-3x} & = & \color{red}{3x} -30 \color{blue}{-3x} \color{blue}{+5} \\\Leftrightarrow & -x& = & -25 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-25}{-1} \\\Leftrightarrow & x = 25 & & \\ & V = \left\{ 25 \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{7}{2}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{273}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+24& = & 273x-312 \\\Leftrightarrow & 26x \color{red}{+24} \color{blue}{-24} \color{blue}{-273x} & = & \color{red}{273x} -312 \color{blue}{-273x} \color{blue}{-24} \\\Leftrightarrow & -247x& = & -336 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-336}{-247} \\\Leftrightarrow & x = \frac{336}{247} & & \\ & V = \left\{ \frac{336}{247} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{7}{2}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 147x-42 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-147x} & = & \color{red}{147x} -42 \color{blue}{-147x} \color{blue}{-12} \\\Leftrightarrow & -133x& = & -54 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-54}{-133} \\\Leftrightarrow & x = \frac{54}{133} & & \\ & V = \left\{ \frac{54}{133} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x+\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 208x+1248 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-208x} & = & \color{red}{208x} +1248 \color{blue}{-208x} \color{blue}{-48} \\\Leftrightarrow & -169x& = & 1200 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{1200}{-169} \\\Leftrightarrow & x = \frac{-1200}{169} & & \\ & V = \left\{ \frac{-1200}{169} \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{-7}{4}x-8 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x-\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+40& = & -245x-1120 \\\Leftrightarrow & 28x \color{red}{+40} \color{blue}{-40} \color{blue}{+245x} & = & \color{red}{-245x} -1120 \color{blue}{+245x} \color{blue}{-40} \\\Leftrightarrow & 273x& = & -1160 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{-1160}{273} \\\Leftrightarrow & x = \frac{-1160}{273} & & \\ & V = \left\{ \frac{-1160}{273} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-336}{ \color{blue}{240} }x-\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -336x-1200 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+336x} & = & \color{red}{-336x} -1200 \color{blue}{+336x} \color{blue}{-45} \\\Leftrightarrow & 376x& = & -1245 \\\Leftrightarrow & \frac{376x}{ \color{red}{376} }& = & \frac{-1245}{376} \\\Leftrightarrow & x = \frac{-1245}{376} & & \\ & V = \left\{ \frac{-1245}{376} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{135}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+20& = & 135x+540 \\\Leftrightarrow & 18x \color{red}{+20} \color{blue}{-20} \color{blue}{-135x} & = & \color{red}{135x} +540 \color{blue}{-135x} \color{blue}{-20} \\\Leftrightarrow & -117x& = & 520 \\\Leftrightarrow & \frac{-117x}{ \color{red}{-117} }& = & \frac{520}{-117} \\\Leftrightarrow & x = \frac{-40}{9} & & \\ & V = \left\{ \frac{-40}{9} \right\} & \\\end{align}\)