Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{5}{6}x-1 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{275}{ \color{blue}{330} }x-\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & 275x-330 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{-275x} & = & \color{red}{275x} -330 \color{blue}{-275x} \color{blue}{-150} \\\Leftrightarrow & -209x& = & -480 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-480}{-209} \\\Leftrightarrow & x = \frac{480}{209} & & \\ & V = \left\{ \frac{480}{209} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{4}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{72}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 72x-90 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-72x} & = & \color{red}{72x} -90 \color{blue}{-72x} \color{blue}{-40} \\\Leftrightarrow & -57x& = & -130 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{-130}{-57} \\\Leftrightarrow & x = \frac{130}{57} & & \\ & V = \left\{ \frac{130}{57} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{15}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 15x-630 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-15x} & = & \color{red}{15x} -630 \color{blue}{-15x} \color{blue}{-40} \\\Leftrightarrow & 3x& = & -670 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-670}{3} \\\Leftrightarrow & x = \frac{-670}{3} & & \\ & V = \left\{ \frac{-670}{3} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 2, 6 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{4}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-25& = & 24x+210 \\\Leftrightarrow & 15x \color{red}{-25} \color{blue}{+25} \color{blue}{-24x} & = & \color{red}{24x} +210 \color{blue}{-24x} \color{blue}{+25} \\\Leftrightarrow & -9x& = & 235 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{235}{-9} \\\Leftrightarrow & x = \frac{-235}{9} & & \\ & V = \left\{ \frac{-235}{9} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 42x-210 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-42x} & = & \color{red}{42x} -210 \color{blue}{-42x} \color{blue}{-120} \\\Leftrightarrow & -7x& = & -330 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-330}{-7} \\\Leftrightarrow & x = \frac{330}{7} & & \\ & V = \left\{ \frac{330}{7} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & 66x-1650 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-66x} & = & \color{red}{66x} -1650 \color{blue}{-66x} \color{blue}{+60} \\\Leftrightarrow & -11x& = & -1590 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-1590}{-11} \\\Leftrightarrow & x = \frac{1590}{11} & & \\ & V = \left\{ \frac{1590}{11} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x+\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & 26x+468 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{-26x} & = & \color{red}{26x} +468 \color{blue}{-26x} \color{blue}{+18} \\\Leftrightarrow & 13x& = & 486 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{486}{13} \\\Leftrightarrow & x = \frac{486}{13} & & \\ & V = \left\{ \frac{486}{13} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -24x-30 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+24x} & = & \color{red}{-24x} -30 \color{blue}{+24x} \color{blue}{+4} \\\Leftrightarrow & 29x& = & -26 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-26}{29} \\\Leftrightarrow & x = \frac{-26}{29} & & \\ & V = \left\{ \frac{-26}{29} \right\} & \\\end{align}\)
9. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{154}{ \color{blue}{110} }x+\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+50& = & 154x+770 \\\Leftrightarrow & 55x \color{red}{+50} \color{blue}{-50} \color{blue}{-154x} & = & \color{red}{154x} +770 \color{blue}{-154x} \color{blue}{-50} \\\Leftrightarrow & -99x& = & 720 \\\Leftrightarrow & \frac{-99x}{ \color{red}{-99} }& = & \frac{720}{-99} \\\Leftrightarrow & x = \frac{-80}{11} & & \\ & V = \left\{ \frac{-80}{11} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{42}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & 42x-420 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{-42x} & = & \color{red}{42x} -420 \color{blue}{-42x} \color{blue}{+35} \\\Leftrightarrow & -30x& = & -385 \\\Leftrightarrow & \frac{-30x}{ \color{red}{-30} }& = & \frac{-385}{-30} \\\Leftrightarrow & x = \frac{77}{6} & & \\ & V = \left\{ \frac{77}{6} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-12& = & 33x+132 \\\Leftrightarrow & 22x \color{red}{-12} \color{blue}{+12} \color{blue}{-33x} & = & \color{red}{33x} +132 \color{blue}{-33x} \color{blue}{+12} \\\Leftrightarrow & -11x& = & 144 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{144}{-11} \\\Leftrightarrow & x = \frac{-144}{11} & & \\ & V = \left\{ \frac{-144}{11} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{2}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{56}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 56x-672 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-56x} & = & \color{red}{56x} -672 \color{blue}{-56x} \color{blue}{+48} \\\Leftrightarrow & -35x& = & -624 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-624}{-35} \\\Leftrightarrow & x = \frac{624}{35} & & \\ & V = \left\{ \frac{624}{35} \right\} & \\\end{align}\)