Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{3}{4}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{45}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-8& = & 45x+360 \\\Leftrightarrow & 20x \color{red}{-8} \color{blue}{+8} \color{blue}{-45x} & = & \color{red}{45x} +360 \color{blue}{-45x} \color{blue}{+8} \\\Leftrightarrow & -25x& = & 368 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{368}{-25} \\\Leftrightarrow & x = \frac{-368}{25} & & \\ & V = \left\{ \frac{-368}{25} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{5}{2}x+7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x+\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 35x+98 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-35x} & = & \color{red}{35x} +98 \color{blue}{-35x} \color{blue}{-4} \\\Leftrightarrow & -33x& = & 94 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{94}{-33} \\\Leftrightarrow & x = \frac{-94}{33} & & \\ & V = \left\{ \frac{-94}{33} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+24& = & 154x+66 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-154x} & = & \color{red}{154x} +66 \color{blue}{-154x} \color{blue}{-24} \\\Leftrightarrow & -121x& = & 42 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{42}{-121} \\\Leftrightarrow & x = \frac{-42}{121} & & \\ & V = \left\{ \frac{-42}{121} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & -264x-1980 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+264x} & = & \color{red}{-264x} -1980 \color{blue}{+264x} \color{blue}{+60} \\\Leftrightarrow & 319x& = & -1920 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{-1920}{319} \\\Leftrightarrow & x = \frac{-1920}{319} & & \\ & V = \left\{ \frac{-1920}{319} \right\} & \\\end{align}\)
5. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-10& = & 6x+108 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{-6x} & = & \color{red}{6x} +108 \color{blue}{-6x} \color{blue}{+10} \\\Leftrightarrow & 3x& = & 118 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{118}{3} \\\Leftrightarrow & x = \frac{118}{3} & & \\ & V = \left\{ \frac{118}{3} \right\} & \\\end{align}\)
6. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{5}{4}x-4 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{60}{ \color{blue}{48} }x-\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+15& = & 60x-192 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-60x} & = & \color{red}{60x} -192 \color{blue}{-60x} \color{blue}{-15} \\\Leftrightarrow & -44x& = & -207 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-207}{-44} \\\Leftrightarrow & x = \frac{207}{44} & & \\ & V = \left\{ \frac{207}{44} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-231}{ \color{blue}{132} }x+\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & -231x+924 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{+231x} & = & \color{red}{-231x} +924 \color{blue}{+231x} \color{blue}{+48} \\\Leftrightarrow & 275x& = & 972 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{972}{275} \\\Leftrightarrow & x = \frac{972}{275} & & \\ & V = \left\{ \frac{972}{275} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x+\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & -88x+396 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{+88x} & = & \color{red}{-88x} +396 \color{blue}{+88x} \color{blue}{-48} \\\Leftrightarrow & 121x& = & 348 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{348}{121} \\\Leftrightarrow & x = \frac{348}{121} & & \\ & V = \left\{ \frac{348}{121} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & -100x+360 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+100x} & = & \color{red}{-100x} +360 \color{blue}{+100x} \color{blue}{+8} \\\Leftrightarrow & 115x& = & 368 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{368}{115} \\\Leftrightarrow & x = \frac{16}{5} & & \\ & V = \left\{ \frac{16}{5} \right\} & \\\end{align}\)
10. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 45 }{ \color{blue}{165} })& = & (\frac{-66}{ \color{blue}{165} }x+\frac{495}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-45& = & -66x+495 \\\Leftrightarrow & 55x \color{red}{-45} \color{blue}{+45} \color{blue}{+66x} & = & \color{red}{-66x} +495 \color{blue}{+66x} \color{blue}{+45} \\\Leftrightarrow & 121x& = & 540 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{540}{121} \\\Leftrightarrow & x = \frac{540}{121} & & \\ & V = \left\{ \frac{540}{121} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x+252 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} +252 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 228 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{228}{-7} \\\Leftrightarrow & x = \frac{-228}{7} & & \\ & V = \left\{ \frac{-228}{7} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{-5}{6}x-3 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x-\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & -75x-270 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{+75x} & = & \color{red}{-75x} -270 \color{blue}{+75x} \color{blue}{-40} \\\Leftrightarrow & 93x& = & -310 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{-310}{93} \\\Leftrightarrow & x = \frac{-10}{3} & & \\ & V = \left\{ \frac{-10}{3} \right\} & \\\end{align}\)