Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 3, 6 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{7}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-25& = & 42x+120 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-42x} & = & \color{red}{42x} +120 \color{blue}{-42x} \color{blue}{+25} \\\Leftrightarrow & -32x& = & 145 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{145}{-32} \\\Leftrightarrow & x = \frac{-145}{32} & & \\ & V = \left\{ \frac{-145}{32} \right\} & \\\end{align}\)
2. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{2}{3}x-4 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 30 }{ \color{blue}{165} })& = & (\frac{110}{ \color{blue}{165} }x-\frac{660}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-30& = & 110x-660 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-110x} & = & \color{red}{110x} -660 \color{blue}{-110x} \color{blue}{+30} \\\Leftrightarrow & -77x& = & -630 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-630}{-77} \\\Leftrightarrow & x = \frac{90}{11} & & \\ & V = \left\{ \frac{90}{11} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & -100x-120 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+100x} & = & \color{red}{-100x} -120 \color{blue}{+100x} \color{blue}{+8} \\\Leftrightarrow & 115x& = & -112 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{-112}{115} \\\Leftrightarrow & x = \frac{-112}{115} & & \\ & V = \left\{ \frac{-112}{115} \right\} & \\\end{align}\)
4. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{20}{ \color{blue}{40} }x+\frac{120}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+25& = & 20x+120 \\\Leftrightarrow & 8x \color{red}{+25} \color{blue}{-25} \color{blue}{-20x} & = & \color{red}{20x} +120 \color{blue}{-20x} \color{blue}{-25} \\\Leftrightarrow & -12x& = & 95 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{95}{-12} \\\Leftrightarrow & x = \frac{-95}{12} & & \\ & V = \left\{ \frac{-95}{12} \right\} & \\\end{align}\)
5. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-24& = & 39x+624 \\\Leftrightarrow & 26x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} +624 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & 648 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{648}{-13} \\\Leftrightarrow & x = \frac{-648}{13} & & \\ & V = \left\{ \frac{-648}{13} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{3}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & 18x-240 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{-18x} & = & \color{red}{18x} -240 \color{blue}{-18x} \color{blue}{+25} \\\Leftrightarrow & -13x& = & -215 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-215}{-13} \\\Leftrightarrow & x = \frac{215}{13} & & \\ & V = \left\{ \frac{215}{13} \right\} & \\\end{align}\)
7. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{5}{2}x+7 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{280}{ \color{blue}{112} }x+\frac{784}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & 280x+784 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{-280x} & = & \color{red}{280x} +784 \color{blue}{-280x} \color{blue}{+35} \\\Leftrightarrow & -264x& = & 819 \\\Leftrightarrow & \frac{-264x}{ \color{red}{-264} }& = & \frac{819}{-264} \\\Leftrightarrow & x = \frac{-273}{88} & & \\ & V = \left\{ \frac{-273}{88} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & -25x+90 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{+25x} & = & \color{red}{-25x} +90 \color{blue}{+25x} \color{blue}{-8} \\\Leftrightarrow & 31x& = & 82 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{82}{31} \\\Leftrightarrow & x = \frac{82}{31} & & \\ & V = \left\{ \frac{82}{31} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x+1470 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} +1470 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & 1530 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{1530}{203} \\\Leftrightarrow & x = \frac{1530}{203} & & \\ & V = \left\{ \frac{1530}{203} \right\} & \\\end{align}\)
10. \(\text{20 is het kleinste gemene veelvoud van 5, 10 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{-7}{4}x-1 \\\Leftrightarrow & \color{blue}{20.} (\frac{4x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{-35}{ \color{blue}{20} }x-\frac{20}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 4x+6& = & -35x-20 \\\Leftrightarrow & 4x \color{red}{+6} \color{blue}{-6} \color{blue}{+35x} & = & \color{red}{-35x} -20 \color{blue}{+35x} \color{blue}{-6} \\\Leftrightarrow & 39x& = & -26 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-26}{39} \\\Leftrightarrow & x = \frac{-2}{3} & & \\ & V = \left\{ \frac{-2}{3} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x-\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 112x-288 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-112x} & = & \color{red}{112x} -288 \color{blue}{-112x} \color{blue}{-15} \\\Leftrightarrow & -88x& = & -303 \\\Leftrightarrow & \frac{-88x}{ \color{red}{-88} }& = & \frac{-303}{-88} \\\Leftrightarrow & x = \frac{303}{88} & & \\ & V = \left\{ \frac{303}{88} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{52}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & 52x+156 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-52x} & = & \color{red}{52x} +156 \color{blue}{-52x} \color{blue}{-24} \\\Leftrightarrow & -13x& = & 132 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{132}{-13} \\\Leftrightarrow & x = \frac{-132}{13} & & \\ & V = \left\{ \frac{-132}{13} \right\} & \\\end{align}\)