Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x-\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+30& = & -52x-260 \\\Leftrightarrow & 65x \color{red}{+30} \color{blue}{-30} \color{blue}{+52x} & = & \color{red}{-52x} -260 \color{blue}{+52x} \color{blue}{-30} \\\Leftrightarrow & 117x& = & -290 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-290}{117} \\\Leftrightarrow & x = \frac{-290}{117} & & \\ & V = \left\{ \frac{-290}{117} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -294x-840 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+294x} & = & \color{red}{-294x} -840 \color{blue}{+294x} \color{blue}{+90} \\\Leftrightarrow & 329x& = & -750 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-750}{329} \\\Leftrightarrow & x = \frac{-750}{329} & & \\ & V = \left\{ \frac{-750}{329} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 30 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x+\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-30& = & -70x+630 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{+70x} & = & \color{red}{-70x} +630 \color{blue}{+70x} \color{blue}{+30} \\\Leftrightarrow & 91x& = & 660 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{660}{91} \\\Leftrightarrow & x = \frac{660}{91} & & \\ & V = \left\{ \frac{660}{91} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & -42x+180 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+42x} & = & \color{red}{-42x} +180 \color{blue}{+42x} \color{blue}{+25} \\\Leftrightarrow & 47x& = & 205 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{205}{47} \\\Leftrightarrow & x = \frac{205}{47} & & \\ & V = \left\{ \frac{205}{47} \right\} & \\\end{align}\)
5. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x+\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 21x+14 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-21x} & = & \color{red}{21x} +14 \color{blue}{-21x} \color{blue}{-4} \\\Leftrightarrow & -19x& = & 10 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{10}{-19} \\\Leftrightarrow & x = \frac{-10}{19} & & \\ & V = \left\{ \frac{-10}{19} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & -156x+3120 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{+156x} & = & \color{red}{-156x} +3120 \color{blue}{+156x} \color{blue}{+60} \\\Leftrightarrow & 221x& = & 3180 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{3180}{221} \\\Leftrightarrow & x = \frac{3180}{221} & & \\ & V = \left\{ \frac{3180}{221} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{5}{6}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{75}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 75x+540 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-75x} & = & \color{red}{75x} +540 \color{blue}{-75x} \color{blue}{+20} \\\Leftrightarrow & -57x& = & 560 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{560}{-57} \\\Leftrightarrow & x = \frac{-560}{57} & & \\ & V = \left\{ \frac{-560}{57} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x+42 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} +42 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & 30 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{30}{-7} \\\Leftrightarrow & x = \frac{-30}{7} & & \\ & V = \left\{ \frac{-30}{7} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 252x-1050 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-252x} & = & \color{red}{252x} -1050 \color{blue}{-252x} \color{blue}{-120} \\\Leftrightarrow & -217x& = & -1170 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-1170}{-217} \\\Leftrightarrow & x = \frac{1170}{217} & & \\ & V = \left\{ \frac{1170}{217} \right\} & \\\end{align}\)
10. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{12}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{6}{ \color{blue}{12} }x-\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-5& = & 6x-84 \\\Leftrightarrow & 4x \color{red}{-5} \color{blue}{+5} \color{blue}{-6x} & = & \color{red}{6x} -84 \color{blue}{-6x} \color{blue}{+5} \\\Leftrightarrow & -2x& = & -79 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-79}{-2} \\\Leftrightarrow & x = \frac{79}{2} & & \\ & V = \left\{ \frac{79}{2} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{2}{3}x+3 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{32}{ \color{blue}{48} }x+\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & 32x+144 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{-32x} & = & \color{red}{32x} +144 \color{blue}{-32x} \color{blue}{+15} \\\Leftrightarrow & -8x& = & 159 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{159}{-8} \\\Leftrightarrow & x = \frac{-159}{8} & & \\ & V = \left\{ \frac{-159}{8} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{11}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 30 }{ \color{blue}{110} })& = & (\frac{-88}{ \color{blue}{110} }x+\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-30& = & -88x+110 \\\Leftrightarrow & 55x \color{red}{-30} \color{blue}{+30} \color{blue}{+88x} & = & \color{red}{-88x} +110 \color{blue}{+88x} \color{blue}{+30} \\\Leftrightarrow & 143x& = & 140 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{140}{143} \\\Leftrightarrow & x = \frac{140}{143} & & \\ & V = \left\{ \frac{140}{143} \right\} & \\\end{align}\)