Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{-176}{ \color{blue}{220} }x-\frac{1540}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-40& = & -176x-1540 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{+176x} & = & \color{red}{-176x} -1540 \color{blue}{+176x} \color{blue}{+40} \\\Leftrightarrow & 231x& = & -1500 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-1500}{231} \\\Leftrightarrow & x = \frac{-500}{77} & & \\ & V = \left\{ \frac{-500}{77} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x-\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & -16x-192 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+16x} & = & \color{red}{-16x} -192 \color{blue}{+16x} \color{blue}{+15} \\\Leftrightarrow & 28x& = & -177 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{-177}{28} \\\Leftrightarrow & x = \frac{-177}{28} & & \\ & V = \left\{ \frac{-177}{28} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x+168 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} +168 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 144 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{144}{-7} \\\Leftrightarrow & x = \frac{-144}{7} & & \\ & V = \left\{ \frac{-144}{7} \right\} & \\\end{align}\)
4. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{5}{4}x-6 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 40 }{ \color{blue}{260} })& = & (\frac{325}{ \color{blue}{260} }x-\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-40& = & 325x-1560 \\\Leftrightarrow & 52x \color{red}{-40} \color{blue}{+40} \color{blue}{-325x} & = & \color{red}{325x} -1560 \color{blue}{-325x} \color{blue}{+40} \\\Leftrightarrow & -273x& = & -1520 \\\Leftrightarrow & \frac{-273x}{ \color{red}{-273} }& = & \frac{-1520}{-273} \\\Leftrightarrow & x = \frac{1520}{273} & & \\ & V = \left\{ \frac{1520}{273} \right\} & \\\end{align}\)
5. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{8}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{56}{ \color{blue}{24} }x+\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+9& = & 56x+72 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{-56x} & = & \color{red}{56x} +72 \color{blue}{-56x} \color{blue}{-9} \\\Leftrightarrow & -44x& = & 63 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{63}{-44} \\\Leftrightarrow & x = \frac{-63}{44} & & \\ & V = \left\{ \frac{-63}{44} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x-\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & 12x-144 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{-12x} & = & \color{red}{12x} -144 \color{blue}{-12x} \color{blue}{+16} \\\Leftrightarrow & -3x& = & -128 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-128}{-3} \\\Leftrightarrow & x = \frac{128}{3} & & \\ & V = \left\{ \frac{128}{3} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 168x-980 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-168x} & = & \color{red}{168x} -980 \color{blue}{-168x} \color{blue}{-80} \\\Leftrightarrow & -133x& = & -1060 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-1060}{-133} \\\Leftrightarrow & x = \frac{1060}{133} & & \\ & V = \left\{ \frac{1060}{133} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{11}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 36 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+36& = & 44x-792 \\\Leftrightarrow & 33x \color{red}{+36} \color{blue}{-36} \color{blue}{-44x} & = & \color{red}{44x} -792 \color{blue}{-44x} \color{blue}{-36} \\\Leftrightarrow & -11x& = & -828 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-828}{-11} \\\Leftrightarrow & x = \frac{828}{11} & & \\ & V = \left\{ \frac{828}{11} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{3}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 18x+180 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-18x} & = & \color{red}{18x} +180 \color{blue}{-18x} \color{blue}{-9} \\\Leftrightarrow & -13x& = & 171 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{171}{-13} \\\Leftrightarrow & x = \frac{-171}{13} & & \\ & V = \left\{ \frac{-171}{13} \right\} & \\\end{align}\)
10. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }+ \frac{ 56 }{ \color{blue}{308} })& = & (\frac{-539}{ \color{blue}{308} }x+\frac{1232}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x+56& = & -539x+1232 \\\Leftrightarrow & 44x \color{red}{+56} \color{blue}{-56} \color{blue}{+539x} & = & \color{red}{-539x} +1232 \color{blue}{+539x} \color{blue}{-56} \\\Leftrightarrow & 583x& = & 1176 \\\Leftrightarrow & \frac{583x}{ \color{red}{583} }& = & \frac{1176}{583} \\\Leftrightarrow & x = \frac{1176}{583} & & \\ & V = \left\{ \frac{1176}{583} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+24& = & 7x-252 \\\Leftrightarrow & 6x \color{red}{+24} \color{blue}{-24} \color{blue}{-7x} & = & \color{red}{7x} -252 \color{blue}{-7x} \color{blue}{-24} \\\Leftrightarrow & -x& = & -276 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-276}{-1} \\\Leftrightarrow & x = 276 & & \\ & V = \left\{ 276 \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -24x+210 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+24x} & = & \color{red}{-24x} +210 \color{blue}{+24x} \color{blue}{+4} \\\Leftrightarrow & 29x& = & 214 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{214}{29} \\\Leftrightarrow & x = \frac{214}{29} & & \\ & V = \left\{ \frac{214}{29} \right\} & \\\end{align}\)