Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 5x+180 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-5x} & = & \color{red}{5x} +180 \color{blue}{-5x} \color{blue}{+4} \\\Leftrightarrow & x& = & 184 \\ & V = \left\{ 184 \right\} & \\\end{align}\)
2. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{2}{ \color{blue}{6} }x+\frac{18}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 2x+18 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-2x} & = & \color{red}{2x} +18 \color{blue}{-2x} \color{blue}{-5} \\\Leftrightarrow & x& = & 13 \\ & V = \left\{ 13 \right\} & \\\end{align}\)
3. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-50& = & 55x-440 \\\Leftrightarrow & 22x \color{red}{-50} \color{blue}{+50} \color{blue}{-55x} & = & \color{red}{55x} -440 \color{blue}{-55x} \color{blue}{+50} \\\Leftrightarrow & -33x& = & -390 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-390}{-33} \\\Leftrightarrow & x = \frac{130}{11} & & \\ & V = \left\{ \frac{130}{11} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-120& = & 65x-1560 \\\Leftrightarrow & 78x \color{red}{-120} \color{blue}{+120} \color{blue}{-65x} & = & \color{red}{65x} -1560 \color{blue}{-65x} \color{blue}{+120} \\\Leftrightarrow & 13x& = & -1440 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-1440}{13} \\\Leftrightarrow & x = \frac{-1440}{13} & & \\ & V = \left\{ \frac{-1440}{13} \right\} & \\\end{align}\)
5. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{104}{ \color{blue}{260} }x+\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-60& = & 104x+2080 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-104x} & = & \color{red}{104x} +2080 \color{blue}{-104x} \color{blue}{+60} \\\Leftrightarrow & -39x& = & 2140 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{2140}{-39} \\\Leftrightarrow & x = \frac{-2140}{39} & & \\ & V = \left\{ \frac{-2140}{39} \right\} & \\\end{align}\)
6. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{7}{2}x+3 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{539}{ \color{blue}{154} }x+\frac{462}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 539x+462 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-539x} & = & \color{red}{539x} +462 \color{blue}{-539x} \color{blue}{-28} \\\Leftrightarrow & -517x& = & 434 \\\Leftrightarrow & \frac{-517x}{ \color{red}{-517} }& = & \frac{434}{-517} \\\Leftrightarrow & x = \frac{-434}{517} & & \\ & V = \left\{ \frac{-434}{517} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-120& = & 35x-1470 \\\Leftrightarrow & 42x \color{red}{-120} \color{blue}{+120} \color{blue}{-35x} & = & \color{red}{35x} -1470 \color{blue}{-35x} \color{blue}{+120} \\\Leftrightarrow & 7x& = & -1350 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1350}{7} \\\Leftrightarrow & x = \frac{-1350}{7} & & \\ & V = \left\{ \frac{-1350}{7} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x+\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 20x+300 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-20x} & = & \color{red}{20x} +300 \color{blue}{-20x} \color{blue}{+18} \\\Leftrightarrow & -5x& = & 318 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{318}{-5} \\\Leftrightarrow & x = \frac{-318}{5} & & \\ & V = \left\{ \frac{-318}{5} \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x+\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-45& = & -84x+630 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{+84x} & = & \color{red}{-84x} +630 \color{blue}{+84x} \color{blue}{+45} \\\Leftrightarrow & 119x& = & 675 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{675}{119} \\\Leftrightarrow & x = \frac{675}{119} & & \\ & V = \left\{ \frac{675}{119} \right\} & \\\end{align}\)
10. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }- \frac{ 4 }{ \color{blue}{15} })& = & (\frac{-10}{ \color{blue}{15} }x+\frac{120}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x-4& = & -10x+120 \\\Leftrightarrow & 3x \color{red}{-4} \color{blue}{+4} \color{blue}{+10x} & = & \color{red}{-10x} +120 \color{blue}{+10x} \color{blue}{+4} \\\Leftrightarrow & 13x& = & 124 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{124}{13} \\\Leftrightarrow & x = \frac{124}{13} & & \\ & V = \left\{ \frac{124}{13} \right\} & \\\end{align}\)
11. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{4}{3}x+4 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 9 }{ \color{blue}{21} })& = & (\frac{28}{ \color{blue}{21} }x+\frac{84}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+9& = & 28x+84 \\\Leftrightarrow & 3x \color{red}{+9} \color{blue}{-9} \color{blue}{-28x} & = & \color{red}{28x} +84 \color{blue}{-28x} \color{blue}{-9} \\\Leftrightarrow & -25x& = & 75 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{75}{-25} \\\Leftrightarrow & x = -3 & & \\ & V = \left\{ -3 \right\} & \\\end{align}\)
12. \(\text{40 is het kleinste gemene veelvoud van 4, 8 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{8}& = & \frac{3}{5}x-7 \\\Leftrightarrow & \color{blue}{40.} (\frac{10x}{ \color{blue}{40} }- \frac{ 15 }{ \color{blue}{40} })& = & (\frac{24}{ \color{blue}{40} }x-\frac{280}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 10x-15& = & 24x-280 \\\Leftrightarrow & 10x \color{red}{-15} \color{blue}{+15} \color{blue}{-24x} & = & \color{red}{24x} -280 \color{blue}{-24x} \color{blue}{+15} \\\Leftrightarrow & -14x& = & -265 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{-265}{-14} \\\Leftrightarrow & x = \frac{265}{14} & & \\ & V = \left\{ \frac{265}{14} \right\} & \\\end{align}\)