Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+15& = & -128x+240 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{+128x} & = & \color{red}{-128x} +240 \color{blue}{+128x} \color{blue}{-15} \\\Leftrightarrow & 140x& = & 225 \\\Leftrightarrow & \frac{140x}{ \color{red}{140} }& = & \frac{225}{140} \\\Leftrightarrow & x = \frac{45}{28} & & \\ & V = \left\{ \frac{45}{28} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-48}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -48x+240 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+48x} & = & \color{red}{-48x} +240 \color{blue}{+48x} \color{blue}{-8} \\\Leftrightarrow & 63x& = & 232 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{232}{63} \\\Leftrightarrow & x = \frac{232}{63} & & \\ & V = \left\{ \frac{232}{63} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-12& = & -130x+156 \\\Leftrightarrow & 39x \color{red}{-12} \color{blue}{+12} \color{blue}{+130x} & = & \color{red}{-130x} +156 \color{blue}{+130x} \color{blue}{+12} \\\Leftrightarrow & 169x& = & 168 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{168}{169} \\\Leftrightarrow & x = \frac{168}{169} & & \\ & V = \left\{ \frac{168}{169} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & -112x+420 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{+112x} & = & \color{red}{-112x} +420 \color{blue}{+112x} \color{blue}{+80} \\\Leftrightarrow & 147x& = & 500 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{500}{147} \\\Leftrightarrow & x = \frac{500}{147} & & \\ & V = \left\{ \frac{500}{147} \right\} & \\\end{align}\)
5. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{3}{5}x-4 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 30 }{ \color{blue}{195} })& = & (\frac{117}{ \color{blue}{195} }x-\frac{780}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+30& = & 117x-780 \\\Leftrightarrow & 65x \color{red}{+30} \color{blue}{-30} \color{blue}{-117x} & = & \color{red}{117x} -780 \color{blue}{-117x} \color{blue}{-30} \\\Leftrightarrow & -52x& = & -810 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{-810}{-52} \\\Leftrightarrow & x = \frac{405}{26} & & \\ & V = \left\{ \frac{405}{26} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{9}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 10 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+10& = & 9x+72 \\\Leftrightarrow & 6x \color{red}{+10} \color{blue}{-10} \color{blue}{-9x} & = & \color{red}{9x} +72 \color{blue}{-9x} \color{blue}{-10} \\\Leftrightarrow & -3x& = & 62 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{62}{-3} \\\Leftrightarrow & x = \frac{-62}{3} & & \\ & V = \left\{ \frac{-62}{3} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-24& = & 33x-528 \\\Leftrightarrow & 22x \color{red}{-24} \color{blue}{+24} \color{blue}{-33x} & = & \color{red}{33x} -528 \color{blue}{-33x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & -504 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-504}{-11} \\\Leftrightarrow & x = \frac{504}{11} & & \\ & V = \left\{ \frac{504}{11} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & 66x+1650 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{-66x} & = & \color{red}{66x} +1650 \color{blue}{-66x} \color{blue}{+120} \\\Leftrightarrow & -11x& = & 1770 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{1770}{-11} \\\Leftrightarrow & x = \frac{-1770}{11} & & \\ & V = \left\{ \frac{-1770}{11} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x+48 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} +48 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & 39 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{39}{-8} \\\Leftrightarrow & x = \frac{-39}{8} & & \\ & V = \left\{ \frac{-39}{8} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & -52x+78 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+52x} & = & \color{red}{-52x} +78 \color{blue}{+52x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & 102 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{102}{91} \\\Leftrightarrow & x = \frac{102}{91} & & \\ & V = \left\{ \frac{102}{91} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{5}{6}x-8 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{200}{ \color{blue}{240} }x-\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & 200x-1920 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{-200x} & = & \color{red}{200x} -1920 \color{blue}{-200x} \color{blue}{-75} \\\Leftrightarrow & -152x& = & -1995 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{-1995}{-152} \\\Leftrightarrow & x = \frac{105}{8} & & \\ & V = \left\{ \frac{105}{8} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -70x-168 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+70x} & = & \color{red}{-70x} -168 \color{blue}{+70x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -192 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-192}{91} \\\Leftrightarrow & x = \frac{-192}{91} & & \\ & V = \left\{ \frac{-192}{91} \right\} & \\\end{align}\)