Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x-\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & -260x-468 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{+260x} & = & \color{red}{-260x} -468 \color{blue}{+260x} \color{blue}{+60} \\\Leftrightarrow & 299x& = & -408 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{-408}{299} \\\Leftrightarrow & x = \frac{-408}{299} & & \\ & V = \left\{ \frac{-408}{299} \right\} & \\\end{align}\)
2. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{9}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+16& = & 9x+108 \\\Leftrightarrow & 12x \color{red}{+16} \color{blue}{-16} \color{blue}{-9x} & = & \color{red}{9x} +108 \color{blue}{-9x} \color{blue}{-16} \\\Leftrightarrow & 3x& = & 92 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{92}{3} \\\Leftrightarrow & x = \frac{92}{3} & & \\ & V = \left\{ \frac{92}{3} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-30& = & 39x+390 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-39x} & = & \color{red}{39x} +390 \color{blue}{-39x} \color{blue}{+30} \\\Leftrightarrow & -13x& = & 420 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{420}{-13} \\\Leftrightarrow & x = \frac{-420}{13} & & \\ & V = \left\{ \frac{-420}{13} \right\} & \\\end{align}\)
4. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{24}{ \color{blue}{120} }x-\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 24x-600 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-24x} & = & \color{red}{24x} -600 \color{blue}{-24x} \color{blue}{+75} \\\Leftrightarrow & -4x& = & -525 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-525}{-4} \\\Leftrightarrow & x = \frac{525}{4} & & \\ & V = \left\{ \frac{525}{4} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-126 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -126 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -102 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-102}{-7} \\\Leftrightarrow & x = \frac{102}{7} & & \\ & V = \left\{ \frac{102}{7} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -70x-336 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+70x} & = & \color{red}{-70x} -336 \color{blue}{+70x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & -354 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-354}{91} \\\Leftrightarrow & x = \frac{-354}{91} & & \\ & V = \left\{ \frac{-354}{91} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 112x+384 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-112x} & = & \color{red}{112x} +384 \color{blue}{-112x} \color{blue}{-9} \\\Leftrightarrow & -88x& = & 375 \\\Leftrightarrow & \frac{-88x}{ \color{red}{-88} }& = & \frac{375}{-88} \\\Leftrightarrow & x = \frac{-375}{88} & & \\ & V = \left\{ \frac{-375}{88} \right\} & \\\end{align}\)
8. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{-275}{ \color{blue}{165} }x-\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-75& = & -275x-330 \\\Leftrightarrow & 33x \color{red}{-75} \color{blue}{+75} \color{blue}{+275x} & = & \color{red}{-275x} -330 \color{blue}{+275x} \color{blue}{+75} \\\Leftrightarrow & 308x& = & -255 \\\Leftrightarrow & \frac{308x}{ \color{red}{308} }& = & \frac{-255}{308} \\\Leftrightarrow & x = \frac{-255}{308} & & \\ & V = \left\{ \frac{-255}{308} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{30}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & 30x+240 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{-30x} & = & \color{red}{30x} +240 \color{blue}{-30x} \color{blue}{-25} \\\Leftrightarrow & -18x& = & 215 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{215}{-18} \\\Leftrightarrow & x = \frac{-215}{18} & & \\ & V = \left\{ \frac{-215}{18} \right\} & \\\end{align}\)
10. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{13}& = & \frac{5}{2}x+1 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 70 }{ \color{blue}{182} })& = & (\frac{455}{ \color{blue}{182} }x+\frac{182}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-70& = & 455x+182 \\\Leftrightarrow & 26x \color{red}{-70} \color{blue}{+70} \color{blue}{-455x} & = & \color{red}{455x} +182 \color{blue}{-455x} \color{blue}{+70} \\\Leftrightarrow & -429x& = & 252 \\\Leftrightarrow & \frac{-429x}{ \color{red}{-429} }& = & \frac{252}{-429} \\\Leftrightarrow & x = \frac{-84}{143} & & \\ & V = \left\{ \frac{-84}{143} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{-7}{4}x-4 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-273}{ \color{blue}{156} }x-\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+48& = & -273x-624 \\\Leftrightarrow & 52x \color{red}{+48} \color{blue}{-48} \color{blue}{+273x} & = & \color{red}{-273x} -624 \color{blue}{+273x} \color{blue}{-48} \\\Leftrightarrow & 325x& = & -672 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{-672}{325} \\\Leftrightarrow & x = \frac{-672}{325} & & \\ & V = \left\{ \frac{-672}{325} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & -112x-980 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+112x} & = & \color{red}{-112x} -980 \color{blue}{+112x} \color{blue}{-40} \\\Leftrightarrow & 147x& = & -1020 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{-1020}{147} \\\Leftrightarrow & x = \frac{-340}{49} & & \\ & V = \left\{ \frac{-340}{49} \right\} & \\\end{align}\)