Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 175 }{ \color{blue}{385} })& = & (\frac{77}{ \color{blue}{385} }x-\frac{1925}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+175& = & 77x-1925 \\\Leftrightarrow & 55x \color{red}{+175} \color{blue}{-175} \color{blue}{-77x} & = & \color{red}{77x} -1925 \color{blue}{-77x} \color{blue}{-175} \\\Leftrightarrow & -22x& = & -2100 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{-2100}{-22} \\\Leftrightarrow & x = \frac{1050}{11} & & \\ & V = \left\{ \frac{1050}{11} \right\} & \\\end{align}\)
2. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 154x+132 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-154x} & = & \color{red}{154x} +132 \color{blue}{-154x} \color{blue}{+30} \\\Leftrightarrow & -121x& = & 162 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{162}{-121} \\\Leftrightarrow & x = \frac{-162}{121} & & \\ & V = \left\{ \frac{-162}{121} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x+540 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} +540 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & 580 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{580}{-27} \\\Leftrightarrow & x = \frac{-580}{27} & & \\ & V = \left\{ \frac{-580}{27} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{64}{ \color{blue}{48} }x+\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-9& = & 64x+48 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{-64x} & = & \color{red}{64x} +48 \color{blue}{-64x} \color{blue}{+9} \\\Leftrightarrow & -52x& = & 57 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{57}{-52} \\\Leftrightarrow & x = \frac{-57}{52} & & \\ & V = \left\{ \frac{-57}{52} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 56x+84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-56x} & = & \color{red}{56x} +84 \color{blue}{-56x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & 108 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{108}{-35} \\\Leftrightarrow & x = \frac{-108}{35} & & \\ & V = \left\{ \frac{-108}{35} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & -260x-780 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{+260x} & = & \color{red}{-260x} -780 \color{blue}{+260x} \color{blue}{+60} \\\Leftrightarrow & 299x& = & -720 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{-720}{299} \\\Leftrightarrow & x = \frac{-720}{299} & & \\ & V = \left\{ \frac{-720}{299} \right\} & \\\end{align}\)
7. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }- \frac{ 2 }{ \color{blue}{15} })& = & (\frac{5}{ \color{blue}{15} }x-\frac{60}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x-2& = & 5x-60 \\\Leftrightarrow & 3x \color{red}{-2} \color{blue}{+2} \color{blue}{-5x} & = & \color{red}{5x} -60 \color{blue}{-5x} \color{blue}{+2} \\\Leftrightarrow & -2x& = & -58 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-58}{-2} \\\Leftrightarrow & x = 29 & & \\ & V = \left\{ 29 \right\} & \\\end{align}\)
8. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 30 }{ \color{blue}{165} })& = & (\frac{33}{ \color{blue}{165} }x+\frac{165}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-30& = & 33x+165 \\\Leftrightarrow & 55x \color{red}{-30} \color{blue}{+30} \color{blue}{-33x} & = & \color{red}{33x} +165 \color{blue}{-33x} \color{blue}{+30} \\\Leftrightarrow & 22x& = & 195 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{195}{22} \\\Leftrightarrow & x = \frac{195}{22} & & \\ & V = \left\{ \frac{195}{22} \right\} & \\\end{align}\)
9. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{8}{ \color{blue}{24} }x-\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & 8x-168 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{-8x} & = & \color{red}{8x} -168 \color{blue}{-8x} \color{blue}{+15} \\\Leftrightarrow & 4x& = & -153 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-153}{4} \\\Leftrightarrow & x = \frac{-153}{4} & & \\ & V = \left\{ \frac{-153}{4} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{5}{4}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{60}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 60x-96 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-60x} & = & \color{red}{60x} -96 \color{blue}{-60x} \color{blue}{-9} \\\Leftrightarrow & -44x& = & -105 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-105}{-44} \\\Leftrightarrow & x = \frac{105}{44} & & \\ & V = \left\{ \frac{105}{44} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 21x-252 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{-18} \\\Leftrightarrow & -7x& = & -270 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-270}{-7} \\\Leftrightarrow & x = \frac{270}{7} & & \\ & V = \left\{ \frac{270}{7} \right\} & \\\end{align}\)
12. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{2}{3}x-2 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 9 }{ \color{blue}{21} })& = & (\frac{14}{ \color{blue}{21} }x-\frac{42}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+9& = & 14x-42 \\\Leftrightarrow & 3x \color{red}{+9} \color{blue}{-9} \color{blue}{-14x} & = & \color{red}{14x} -42 \color{blue}{-14x} \color{blue}{-9} \\\Leftrightarrow & -11x& = & -51 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-51}{-11} \\\Leftrightarrow & x = \frac{51}{11} & & \\ & V = \left\{ \frac{51}{11} \right\} & \\\end{align}\)