Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}-\frac{5}{14}=\frac{7}{2}x+5\)
2. \(\frac{x}{3}-\frac{5}{16}=\frac{1}{2}x+4\)
3. \(\frac{x}{2}-\frac{4}{9}=\frac{-2}{3}x+7\)
4. \(\frac{x}{6}+\frac{2}{15}=\frac{6}{5}x+5\)
5. \(\frac{x}{4}-\frac{5}{12}=\frac{2}{3}x-7\)
6. \(\frac{x}{4}-\frac{5}{14}=\frac{-7}{5}x+5\)
7. \(\frac{x}{5}-\frac{4}{15}=\frac{-3}{4}x-6\)
8. \(\frac{x}{3}-\frac{4}{9}=\frac{3}{5}x+6\)
9. \(\frac{x}{7}-\frac{4}{11}=\frac{1}{4}x+3\)
10. \(\frac{x}{2}+\frac{3}{13}=\frac{1}{3}x+2\)
11. \(\frac{x}{6}+\frac{5}{11}=\frac{-2}{5}x-6\)
12. \(\frac{x}{2}+\frac{4}{15}=\frac{2}{5}x-3\)

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{14}& = & \frac{7}{2}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 25 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-25& = & 245x+350 \\\Leftrightarrow & 14x \color{red}{-25} \color{blue}{+25} \color{blue}{-245x} & = & \color{red}{245x} +350 \color{blue}{-245x} \color{blue}{+25} \\\Leftrightarrow & -231x& = & 375 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{375}{-231} \\\Leftrightarrow & x = \frac{-125}{77} & & \\ & V = \left\{ \frac{-125}{77} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & 24x+192 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-24x} & = & \color{red}{24x} +192 \color{blue}{-24x} \color{blue}{+15} \\\Leftrightarrow & -8x& = & 207 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{207}{-8} \\\Leftrightarrow & x = \frac{-207}{8} & & \\ & V = \left\{ \frac{-207}{8} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x+\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & -12x+126 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+12x} & = & \color{red}{-12x} +126 \color{blue}{+12x} \color{blue}{+8} \\\Leftrightarrow & 21x& = & 134 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{134}{21} \\\Leftrightarrow & x = \frac{134}{21} & & \\ & V = \left\{ \frac{134}{21} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 36x+150 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-36x} & = & \color{red}{36x} +150 \color{blue}{-36x} \color{blue}{-4} \\\Leftrightarrow & -31x& = & 146 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{146}{-31} \\\Leftrightarrow & x = \frac{-146}{31} & & \\ & V = \left\{ \frac{-146}{31} \right\} & \\\end{align}\)
5. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{2}{3}x-7 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{8}{ \color{blue}{12} }x-\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-5& = & 8x-84 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-8x} & = & \color{red}{8x} -84 \color{blue}{-8x} \color{blue}{+5} \\\Leftrightarrow & -5x& = & -79 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-79}{-5} \\\Leftrightarrow & x = \frac{79}{5} & & \\ & V = \left\{ \frac{79}{5} \right\} & \\\end{align}\)
6. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{14}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 50 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-50& = & -196x+700 \\\Leftrightarrow & 35x \color{red}{-50} \color{blue}{+50} \color{blue}{+196x} & = & \color{red}{-196x} +700 \color{blue}{+196x} \color{blue}{+50} \\\Leftrightarrow & 231x& = & 750 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{750}{231} \\\Leftrightarrow & x = \frac{250}{77} & & \\ & V = \left\{ \frac{250}{77} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x-\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-16& = & -45x-360 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{+45x} & = & \color{red}{-45x} -360 \color{blue}{+45x} \color{blue}{+16} \\\Leftrightarrow & 57x& = & -344 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{-344}{57} \\\Leftrightarrow & x = \frac{-344}{57} & & \\ & V = \left\{ \frac{-344}{57} \right\} & \\\end{align}\)
8. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{3}{5}x+6 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{27}{ \color{blue}{45} }x+\frac{270}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x-20& = & 27x+270 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-27x} & = & \color{red}{27x} +270 \color{blue}{-27x} \color{blue}{+20} \\\Leftrightarrow & -12x& = & 290 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{290}{-12} \\\Leftrightarrow & x = \frac{-145}{6} & & \\ & V = \left\{ \frac{-145}{6} \right\} & \\\end{align}\)
9. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }- \frac{ 112 }{ \color{blue}{308} })& = & (\frac{77}{ \color{blue}{308} }x+\frac{924}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x-112& = & 77x+924 \\\Leftrightarrow & 44x \color{red}{-112} \color{blue}{+112} \color{blue}{-77x} & = & \color{red}{77x} +924 \color{blue}{-77x} \color{blue}{+112} \\\Leftrightarrow & -33x& = & 1036 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{1036}{-33} \\\Leftrightarrow & x = \frac{-1036}{33} & & \\ & V = \left\{ \frac{-1036}{33} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & 26x+156 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{-26x} & = & \color{red}{26x} +156 \color{blue}{-26x} \color{blue}{-18} \\\Leftrightarrow & 13x& = & 138 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{138}{13} \\\Leftrightarrow & x = \frac{138}{13} & & \\ & V = \left\{ \frac{138}{13} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & -132x-1980 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{+132x} & = & \color{red}{-132x} -1980 \color{blue}{+132x} \color{blue}{-150} \\\Leftrightarrow & 187x& = & -2130 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-2130}{187} \\\Leftrightarrow & x = \frac{-2130}{187} & & \\ & V = \left\{ \frac{-2130}{187} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{2}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 12x-90 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-12x} & = & \color{red}{12x} -90 \color{blue}{-12x} \color{blue}{-8} \\\Leftrightarrow & 3x& = & -98 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-98}{3} \\\Leftrightarrow & x = \frac{-98}{3} & & \\ & V = \left\{ \frac{-98}{3} \right\} & \\\end{align}\)