Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 140x+588 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-140x} & = & \color{red}{140x} +588 \color{blue}{-140x} \color{blue}{-24} \\\Leftrightarrow & -119x& = & 564 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{564}{-119} \\\Leftrightarrow & x = \frac{-564}{119} & & \\ & V = \left\{ \frac{-564}{119} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{8}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{32}{ \color{blue}{24} }x+\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+9& = & 32x+168 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{-32x} & = & \color{red}{32x} +168 \color{blue}{-32x} \color{blue}{-9} \\\Leftrightarrow & -20x& = & 159 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{159}{-20} \\\Leftrightarrow & x = \frac{-159}{20} & & \\ & V = \left\{ \frac{-159}{20} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x+54 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} +54 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & 50 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{50}{-3} \\\Leftrightarrow & x = \frac{-50}{3} & & \\ & V = \left\{ \frac{-50}{3} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{7}{6}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & 245x-630 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{-245x} & = & \color{red}{245x} -630 \color{blue}{-245x} \color{blue}{-28} \\\Leftrightarrow & -215x& = & -658 \\\Leftrightarrow & \frac{-215x}{ \color{red}{-215} }& = & \frac{-658}{-215} \\\Leftrightarrow & x = \frac{658}{215} & & \\ & V = \left\{ \frac{658}{215} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 6x-120 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} -120 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & -x& = & -116 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-116}{-1} \\\Leftrightarrow & x = 116 & & \\ & V = \left\{ 116 \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & 70x-210 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{-70x} & = & \color{red}{70x} -210 \color{blue}{-70x} \color{blue}{-4} \\\Leftrightarrow & -55x& = & -214 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-214}{-55} \\\Leftrightarrow & x = \frac{214}{55} & & \\ & V = \left\{ \frac{214}{55} \right\} & \\\end{align}\)
7. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{-18}{ \color{blue}{45} }x-\frac{45}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x-20& = & -18x-45 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{+18x} & = & \color{red}{-18x} -45 \color{blue}{+18x} \color{blue}{+20} \\\Leftrightarrow & 33x& = & -25 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-25}{33} \\\Leftrightarrow & x = \frac{-25}{33} & & \\ & V = \left\{ \frac{-25}{33} \right\} & \\\end{align}\)
8. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 12 }{ \color{blue}{21} })& = & (\frac{49}{ \color{blue}{21} }x+\frac{42}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+12& = & 49x+42 \\\Leftrightarrow & 3x \color{red}{+12} \color{blue}{-12} \color{blue}{-49x} & = & \color{red}{49x} +42 \color{blue}{-49x} \color{blue}{-12} \\\Leftrightarrow & -46x& = & 30 \\\Leftrightarrow & \frac{-46x}{ \color{red}{-46} }& = & \frac{30}{-46} \\\Leftrightarrow & x = \frac{-15}{23} & & \\ & V = \left\{ \frac{-15}{23} \right\} & \\\end{align}\)
9. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{12}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{4}{ \color{blue}{12} }x-\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x-5& = & 4x-36 \\\Leftrightarrow & 6x \color{red}{-5} \color{blue}{+5} \color{blue}{-4x} & = & \color{red}{4x} -36 \color{blue}{-4x} \color{blue}{+5} \\\Leftrightarrow & 2x& = & -31 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{-31}{2} \\\Leftrightarrow & x = \frac{-31}{2} & & \\ & V = \left\{ \frac{-31}{2} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{5}{6}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 25x-150 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-25x} & = & \color{red}{25x} -150 \color{blue}{-25x} \color{blue}{-8} \\\Leftrightarrow & -19x& = & -158 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-158}{-19} \\\Leftrightarrow & x = \frac{158}{19} & & \\ & V = \left\{ \frac{158}{19} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x-\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-60& = & -117x-936 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{+117x} & = & \color{red}{-117x} -936 \color{blue}{+117x} \color{blue}{+60} \\\Leftrightarrow & 169x& = & -876 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-876}{169} \\\Leftrightarrow & x = \frac{-876}{169} & & \\ & V = \left\{ \frac{-876}{169} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -88x-396 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+88x} & = & \color{red}{-88x} -396 \color{blue}{+88x} \color{blue}{-24} \\\Leftrightarrow & 121x& = & -420 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-420}{121} \\\Leftrightarrow & x = \frac{-420}{121} & & \\ & V = \left\{ \frac{-420}{121} \right\} & \\\end{align}\)