Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{-7}{4}x+3 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 56 }{ \color{blue}{364} })& = & (\frac{-637}{ \color{blue}{364} }x+\frac{1092}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-56& = & -637x+1092 \\\Leftrightarrow & 52x \color{red}{-56} \color{blue}{+56} \color{blue}{+637x} & = & \color{red}{-637x} +1092 \color{blue}{+637x} \color{blue}{+56} \\\Leftrightarrow & 689x& = & 1148 \\\Leftrightarrow & \frac{689x}{ \color{red}{689} }& = & \frac{1148}{689} \\\Leftrightarrow & x = \frac{1148}{689} & & \\ & V = \left\{ \frac{1148}{689} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 168x+420 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-168x} & = & \color{red}{168x} +420 \color{blue}{-168x} \color{blue}{+60} \\\Leftrightarrow & -133x& = & 480 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{480}{-133} \\\Leftrightarrow & x = \frac{-480}{133} & & \\ & V = \left\{ \frac{-480}{133} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 14x+168 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-14x} & = & \color{red}{14x} +168 \color{blue}{-14x} \color{blue}{+35} \\\Leftrightarrow & -8x& = & 203 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{203}{-8} \\\Leftrightarrow & x = \frac{-203}{8} & & \\ & V = \left\{ \frac{-203}{8} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{3}{5}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{198}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 198x+1980 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-198x} & = & \color{red}{198x} +1980 \color{blue}{-198x} \color{blue}{+150} \\\Leftrightarrow & -143x& = & 2130 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{2130}{-143} \\\Leftrightarrow & x = \frac{-2130}{143} & & \\ & V = \left\{ \frac{-2130}{143} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{21}{ \color{blue}{105} }x+\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & 21x+105 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{-21x} & = & \color{red}{21x} +105 \color{blue}{-21x} \color{blue}{+14} \\\Leftrightarrow & -6x& = & 119 \\\Leftrightarrow & \frac{-6x}{ \color{red}{-6} }& = & \frac{119}{-6} \\\Leftrightarrow & x = \frac{-119}{6} & & \\ & V = \left\{ \frac{-119}{6} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{27}{ \color{blue}{18} }x+\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 27x+18 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-27x} & = & \color{red}{27x} +18 \color{blue}{-27x} \color{blue}{-4} \\\Leftrightarrow & -21x& = & 14 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{14}{-21} \\\Leftrightarrow & x = \frac{-2}{3} & & \\ & V = \left\{ \frac{-2}{3} \right\} & \\\end{align}\)
7. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x+\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & 12x+180 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} +180 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & 188 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{188}{-3} \\\Leftrightarrow & x = \frac{-188}{3} & & \\ & V = \left\{ \frac{-188}{3} \right\} & \\\end{align}\)
8. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{1}{4}x-4 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{45}{ \color{blue}{180} }x-\frac{720}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x+40& = & 45x-720 \\\Leftrightarrow & 36x \color{red}{+40} \color{blue}{-40} \color{blue}{-45x} & = & \color{red}{45x} -720 \color{blue}{-45x} \color{blue}{-40} \\\Leftrightarrow & -9x& = & -760 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-760}{-9} \\\Leftrightarrow & x = \frac{760}{9} & & \\ & V = \left\{ \frac{760}{9} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & -25x-210 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{+25x} & = & \color{red}{-25x} -210 \color{blue}{+25x} \color{blue}{+4} \\\Leftrightarrow & 31x& = & -206 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{-206}{31} \\\Leftrightarrow & x = \frac{-206}{31} & & \\ & V = \left\{ \frac{-206}{31} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+4& = & 15x+90 \\\Leftrightarrow & 10x \color{red}{+4} \color{blue}{-4} \color{blue}{-15x} & = & \color{red}{15x} +90 \color{blue}{-15x} \color{blue}{-4} \\\Leftrightarrow & -5x& = & 86 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{86}{-5} \\\Leftrightarrow & x = \frac{-86}{5} & & \\ & V = \left\{ \frac{-86}{5} \right\} & \\\end{align}\)
11. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{2}{ \color{blue}{6} }x-\frac{36}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 2x-36 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-2x} & = & \color{red}{2x} -36 \color{blue}{-2x} \color{blue}{-5} \\\Leftrightarrow & x& = & -41 \\ & V = \left\{ -41 \right\} & \\\end{align}\)
12. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{165}{ \color{blue}{66} }x-\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+12& = & 165x-528 \\\Leftrightarrow & 22x \color{red}{+12} \color{blue}{-12} \color{blue}{-165x} & = & \color{red}{165x} -528 \color{blue}{-165x} \color{blue}{-12} \\\Leftrightarrow & -143x& = & -540 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-540}{-143} \\\Leftrightarrow & x = \frac{540}{143} & & \\ & V = \left\{ \frac{540}{143} \right\} & \\\end{align}\)