Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-18& = & -45x-300 \\\Leftrightarrow & 20x \color{red}{-18} \color{blue}{+18} \color{blue}{+45x} & = & \color{red}{-45x} -300 \color{blue}{+45x} \color{blue}{+18} \\\Leftrightarrow & 65x& = & -282 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{-282}{65} \\\Leftrightarrow & x = \frac{-282}{65} & & \\ & V = \left\{ \frac{-282}{65} \right\} & \\\end{align}\)
2. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 42 }{ \color{blue}{231} })& = & (\frac{77}{ \color{blue}{231} }x-\frac{924}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-42& = & 77x-924 \\\Leftrightarrow & 33x \color{red}{-42} \color{blue}{+42} \color{blue}{-77x} & = & \color{red}{77x} -924 \color{blue}{-77x} \color{blue}{+42} \\\Leftrightarrow & -44x& = & -882 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-882}{-44} \\\Leftrightarrow & x = \frac{441}{22} & & \\ & V = \left\{ \frac{441}{22} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-18& = & 33x+198 \\\Leftrightarrow & 22x \color{red}{-18} \color{blue}{+18} \color{blue}{-33x} & = & \color{red}{33x} +198 \color{blue}{-33x} \color{blue}{+18} \\\Leftrightarrow & -11x& = & 216 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{216}{-11} \\\Leftrightarrow & x = \frac{-216}{11} & & \\ & V = \left\{ \frac{-216}{11} \right\} & \\\end{align}\)
4. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }+ \frac{ 60 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x+\frac{1300}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x+60& = & 65x+1300 \\\Leftrightarrow & 52x \color{red}{+60} \color{blue}{-60} \color{blue}{-65x} & = & \color{red}{65x} +1300 \color{blue}{-65x} \color{blue}{-60} \\\Leftrightarrow & -13x& = & 1240 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{1240}{-13} \\\Leftrightarrow & x = \frac{-1240}{13} & & \\ & V = \left\{ \frac{-1240}{13} \right\} & \\\end{align}\)
5. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 80 }{ \color{blue}{180} })& = & (\frac{216}{ \color{blue}{180} }x-\frac{360}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-80& = & 216x-360 \\\Leftrightarrow & 45x \color{red}{-80} \color{blue}{+80} \color{blue}{-216x} & = & \color{red}{216x} -360 \color{blue}{-216x} \color{blue}{+80} \\\Leftrightarrow & -171x& = & -280 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{-280}{-171} \\\Leftrightarrow & x = \frac{280}{171} & & \\ & V = \left\{ \frac{280}{171} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x-210 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & -198 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-198}{-7} \\\Leftrightarrow & x = \frac{198}{7} & & \\ & V = \left\{ \frac{198}{7} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{2}{3}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{28}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 28x-84 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-28x} & = & \color{red}{28x} -84 \color{blue}{-28x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & -72 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-72}{-7} \\\Leftrightarrow & x = \frac{72}{7} & & \\ & V = \left\{ \frac{72}{7} \right\} & \\\end{align}\)
8. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{7}{4}x+3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 8 }{ \color{blue}{28} })& = & (\frac{49}{ \color{blue}{28} }x+\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+8& = & 49x+84 \\\Leftrightarrow & 4x \color{red}{+8} \color{blue}{-8} \color{blue}{-49x} & = & \color{red}{49x} +84 \color{blue}{-49x} \color{blue}{-8} \\\Leftrightarrow & -45x& = & 76 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{76}{-45} \\\Leftrightarrow & x = \frac{-76}{45} & & \\ & V = \left\{ \frac{-76}{45} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 28x+336 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-28x} & = & \color{red}{28x} +336 \color{blue}{-28x} \color{blue}{-48} \\\Leftrightarrow & -7x& = & 288 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{288}{-7} \\\Leftrightarrow & x = \frac{-288}{7} & & \\ & V = \left\{ \frac{-288}{7} \right\} & \\\end{align}\)
10. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{4}{5}x+5 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 100 }{ \color{blue}{220} })& = & (\frac{176}{ \color{blue}{220} }x+\frac{1100}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-100& = & 176x+1100 \\\Leftrightarrow & 55x \color{red}{-100} \color{blue}{+100} \color{blue}{-176x} & = & \color{red}{176x} +1100 \color{blue}{-176x} \color{blue}{+100} \\\Leftrightarrow & -121x& = & 1200 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{1200}{-121} \\\Leftrightarrow & x = \frac{-1200}{121} & & \\ & V = \left\{ \frac{-1200}{121} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 2, 12 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{12}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{30x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-48}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 30x+25& = & -48x-420 \\\Leftrightarrow & 30x \color{red}{+25} \color{blue}{-25} \color{blue}{+48x} & = & \color{red}{-48x} -420 \color{blue}{+48x} \color{blue}{-25} \\\Leftrightarrow & 78x& = & -445 \\\Leftrightarrow & \frac{78x}{ \color{red}{78} }& = & \frac{-445}{78} \\\Leftrightarrow & x = \frac{-445}{78} & & \\ & V = \left\{ \frac{-445}{78} \right\} & \\\end{align}\)
12. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 12 }{ \color{blue}{21} })& = & (\frac{-14}{ \color{blue}{21} }x-\frac{126}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-12& = & -14x-126 \\\Leftrightarrow & 3x \color{red}{-12} \color{blue}{+12} \color{blue}{+14x} & = & \color{red}{-14x} -126 \color{blue}{+14x} \color{blue}{+12} \\\Leftrightarrow & 17x& = & -114 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-114}{17} \\\Leftrightarrow & x = \frac{-114}{17} & & \\ & V = \left\{ \frac{-114}{17} \right\} & \\\end{align}\)