Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{10}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & 10x-420 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{-10x} & = & \color{red}{10x} -420 \color{blue}{-10x} \color{blue}{+25} \\\Leftrightarrow & 2x& = & -395 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{-395}{2} \\\Leftrightarrow & x = \frac{-395}{2} & & \\ & V = \left\{ \frac{-395}{2} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{5}{6}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 63 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+63& = & 175x+840 \\\Leftrightarrow & 30x \color{red}{+63} \color{blue}{-63} \color{blue}{-175x} & = & \color{red}{175x} +840 \color{blue}{-175x} \color{blue}{-63} \\\Leftrightarrow & -145x& = & 777 \\\Leftrightarrow & \frac{-145x}{ \color{red}{-145} }& = & \frac{777}{-145} \\\Leftrightarrow & x = \frac{-777}{145} & & \\ & V = \left\{ \frac{-777}{145} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-2}{5}x-4 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-32}{ \color{blue}{80} }x-\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+25& = & -32x-320 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{+32x} & = & \color{red}{-32x} -320 \color{blue}{+32x} \color{blue}{-25} \\\Leftrightarrow & 52x& = & -345 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{-345}{52} \\\Leftrightarrow & x = \frac{-345}{52} & & \\ & V = \left\{ \frac{-345}{52} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & -112x+840 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+112x} & = & \color{red}{-112x} +840 \color{blue}{+112x} \color{blue}{+40} \\\Leftrightarrow & 147x& = & 880 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{880}{147} \\\Leftrightarrow & x = \frac{880}{147} & & \\ & V = \left\{ \frac{880}{147} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{1}{4}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{12}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & 12x-240 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-12x} & = & \color{red}{12x} -240 \color{blue}{-12x} \color{blue}{+15} \\\Leftrightarrow & 4x& = & -225 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-225}{4} \\\Leftrightarrow & x = \frac{-225}{4} & & \\ & V = \left\{ \frac{-225}{4} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{336}{ \color{blue}{240} }x+\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x-45& = & 336x+1680 \\\Leftrightarrow & 80x \color{red}{-45} \color{blue}{+45} \color{blue}{-336x} & = & \color{red}{336x} +1680 \color{blue}{-336x} \color{blue}{+45} \\\Leftrightarrow & -256x& = & 1725 \\\Leftrightarrow & \frac{-256x}{ \color{red}{-256} }& = & \frac{1725}{-256} \\\Leftrightarrow & x = \frac{-1725}{256} & & \\ & V = \left\{ \frac{-1725}{256} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 50 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+50& = & 72x+360 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{-72x} & = & \color{red}{72x} +360 \color{blue}{-72x} \color{blue}{-50} \\\Leftrightarrow & -57x& = & 310 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{310}{-57} \\\Leftrightarrow & x = \frac{-310}{57} & & \\ & V = \left\{ \frac{-310}{57} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+60& = & 33x-396 \\\Leftrightarrow & 44x \color{red}{+60} \color{blue}{-60} \color{blue}{-33x} & = & \color{red}{33x} -396 \color{blue}{-33x} \color{blue}{-60} \\\Leftrightarrow & 11x& = & -456 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-456}{11} \\\Leftrightarrow & x = \frac{-456}{11} & & \\ & V = \left\{ \frac{-456}{11} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 15x-30 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} -30 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -9x& = & -22 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-22}{-9} \\\Leftrightarrow & x = \frac{22}{9} & & \\ & V = \left\{ \frac{22}{9} \right\} & \\\end{align}\)
10. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 60 }{ \color{blue}{165} })& = & (\frac{55}{ \color{blue}{165} }x+\frac{660}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-60& = & 55x+660 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{-55x} & = & \color{red}{55x} +660 \color{blue}{-55x} \color{blue}{+60} \\\Leftrightarrow & -22x& = & 720 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{720}{-22} \\\Leftrightarrow & x = \frac{-360}{11} & & \\ & V = \left\{ \frac{-360}{11} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{45}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 45x+30 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-45x} & = & \color{red}{45x} +30 \color{blue}{-45x} \color{blue}{+4} \\\Leftrightarrow & -35x& = & 34 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{34}{-35} \\\Leftrightarrow & x = \frac{-34}{35} & & \\ & V = \left\{ \frac{-34}{35} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -112x+84 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+112x} & = & \color{red}{-112x} +84 \color{blue}{+112x} \color{blue}{-12} \\\Leftrightarrow & 133x& = & 72 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{72}{133} \\\Leftrightarrow & x = \frac{72}{133} & & \\ & V = \left\{ \frac{72}{133} \right\} & \\\end{align}\)