Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{3}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 63x-294 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-63x} & = & \color{red}{63x} -294 \color{blue}{-63x} \color{blue}{-24} \\\Leftrightarrow & -49x& = & -318 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-318}{-49} \\\Leftrightarrow & x = \frac{318}{49} & & \\ & V = \left\{ \frac{318}{49} \right\} & \\\end{align}\)
2. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{7}{2}x+5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{280}{ \color{blue}{80} }x+\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 280x+400 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-280x} & = & \color{red}{280x} +400 \color{blue}{-280x} \color{blue}{+25} \\\Leftrightarrow & -264x& = & 425 \\\Leftrightarrow & \frac{-264x}{ \color{red}{-264} }& = & \frac{425}{-264} \\\Leftrightarrow & x = \frac{-425}{264} & & \\ & V = \left\{ \frac{-425}{264} \right\} & \\\end{align}\)
3. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{11}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & -132x+990 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{+132x} & = & \color{red}{-132x} +990 \color{blue}{+132x} \color{blue}{-60} \\\Leftrightarrow & 187x& = & 930 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{930}{187} \\\Leftrightarrow & x = \frac{930}{187} & & \\ & V = \left\{ \frac{930}{187} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{-7}{4}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-273}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & -273x+156 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{+273x} & = & \color{red}{-273x} +156 \color{blue}{+273x} \color{blue}{+36} \\\Leftrightarrow & 325x& = & 192 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{192}{325} \\\Leftrightarrow & x = \frac{192}{325} & & \\ & V = \left\{ \frac{192}{325} \right\} & \\\end{align}\)
5. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{5}{3}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{130}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & 130x-390 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-130x} & = & \color{red}{130x} -390 \color{blue}{-130x} \color{blue}{+24} \\\Leftrightarrow & -91x& = & -366 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-366}{-91} \\\Leftrightarrow & x = \frac{366}{91} & & \\ & V = \left\{ \frac{366}{91} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{105}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+8& = & 105x+210 \\\Leftrightarrow & 10x \color{red}{+8} \color{blue}{-8} \color{blue}{-105x} & = & \color{red}{105x} +210 \color{blue}{-105x} \color{blue}{-8} \\\Leftrightarrow & -95x& = & 202 \\\Leftrightarrow & \frac{-95x}{ \color{red}{-95} }& = & \frac{202}{-95} \\\Leftrightarrow & x = \frac{-202}{95} & & \\ & V = \left\{ \frac{-202}{95} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{14}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 15 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-15& = & 21x+42 \\\Leftrightarrow & 14x \color{red}{-15} \color{blue}{+15} \color{blue}{-21x} & = & \color{red}{21x} +42 \color{blue}{-21x} \color{blue}{+15} \\\Leftrightarrow & -7x& = & 57 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{57}{-7} \\\Leftrightarrow & x = \frac{-57}{7} & & \\ & V = \left\{ \frac{-57}{7} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x+\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & -176x+264 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+176x} & = & \color{red}{-176x} +264 \color{blue}{+176x} \color{blue}{-30} \\\Leftrightarrow & 209x& = & 234 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{234}{209} \\\Leftrightarrow & x = \frac{234}{209} & & \\ & V = \left\{ \frac{234}{209} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-24& = & 39x-156 \\\Leftrightarrow & 26x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} -156 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & -132 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-132}{-13} \\\Leftrightarrow & x = \frac{132}{13} & & \\ & V = \left\{ \frac{132}{13} \right\} & \\\end{align}\)
10. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{5}{3}x+4 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }- \frac{ 2 }{ \color{blue}{15} })& = & (\frac{25}{ \color{blue}{15} }x+\frac{60}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x-2& = & 25x+60 \\\Leftrightarrow & 3x \color{red}{-2} \color{blue}{+2} \color{blue}{-25x} & = & \color{red}{25x} +60 \color{blue}{-25x} \color{blue}{+2} \\\Leftrightarrow & -22x& = & 62 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{62}{-22} \\\Leftrightarrow & x = \frac{-31}{11} & & \\ & V = \left\{ \frac{-31}{11} \right\} & \\\end{align}\)
11. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 70 }{ \color{blue}{385} })& = & (\frac{539}{ \color{blue}{385} }x+\frac{2695}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-70& = & 539x+2695 \\\Leftrightarrow & 55x \color{red}{-70} \color{blue}{+70} \color{blue}{-539x} & = & \color{red}{539x} +2695 \color{blue}{-539x} \color{blue}{+70} \\\Leftrightarrow & -484x& = & 2765 \\\Leftrightarrow & \frac{-484x}{ \color{red}{-484} }& = & \frac{2765}{-484} \\\Leftrightarrow & x = \frac{-2765}{484} & & \\ & V = \left\{ \frac{-2765}{484} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-30& = & -52x+78 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{+52x} & = & \color{red}{-52x} +78 \color{blue}{+52x} \color{blue}{+30} \\\Leftrightarrow & 91x& = & 108 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{108}{91} \\\Leftrightarrow & x = \frac{108}{91} & & \\ & V = \left\{ \frac{108}{91} \right\} & \\\end{align}\)