Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{5}{4}x+2 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{45}{ \color{blue}{36} }x+\frac{72}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-20& = & 45x+72 \\\Leftrightarrow & 12x \color{red}{-20} \color{blue}{+20} \color{blue}{-45x} & = & \color{red}{45x} +72 \color{blue}{-45x} \color{blue}{+20} \\\Leftrightarrow & -33x& = & 92 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{92}{-33} \\\Leftrightarrow & x = \frac{-92}{33} & & \\ & V = \left\{ \frac{-92}{33} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & 70x+210 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{-70x} & = & \color{red}{70x} +210 \color{blue}{-70x} \color{blue}{-4} \\\Leftrightarrow & -55x& = & 206 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{206}{-55} \\\Leftrightarrow & x = \frac{-206}{55} & & \\ & V = \left\{ \frac{-206}{55} \right\} & \\\end{align}\)
3. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 140 }{ \color{blue}{455} })& = & (\frac{91}{ \color{blue}{455} }x+\frac{2730}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+140& = & 91x+2730 \\\Leftrightarrow & 65x \color{red}{+140} \color{blue}{-140} \color{blue}{-91x} & = & \color{red}{91x} +2730 \color{blue}{-91x} \color{blue}{-140} \\\Leftrightarrow & -26x& = & 2590 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{2590}{-26} \\\Leftrightarrow & x = \frac{-1295}{13} & & \\ & V = \left\{ \frac{-1295}{13} \right\} & \\\end{align}\)
4. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }+ \frac{ 10 }{ \color{blue}{45} })& = & (\frac{-30}{ \color{blue}{45} }x+\frac{225}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x+10& = & -30x+225 \\\Leftrightarrow & 9x \color{red}{+10} \color{blue}{-10} \color{blue}{+30x} & = & \color{red}{-30x} +225 \color{blue}{+30x} \color{blue}{-10} \\\Leftrightarrow & 39x& = & 215 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{215}{39} \\\Leftrightarrow & x = \frac{215}{39} & & \\ & V = \left\{ \frac{215}{39} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 45 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x+\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+45& = & -70x+630 \\\Leftrightarrow & 21x \color{red}{+45} \color{blue}{-45} \color{blue}{+70x} & = & \color{red}{-70x} +630 \color{blue}{+70x} \color{blue}{-45} \\\Leftrightarrow & 91x& = & 585 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{585}{91} \\\Leftrightarrow & x = \frac{45}{7} & & \\ & V = \left\{ \frac{45}{7} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -28x+336 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+28x} & = & \color{red}{-28x} +336 \color{blue}{+28x} \color{blue}{-24} \\\Leftrightarrow & 49x& = & 312 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{312}{49} \\\Leftrightarrow & x = \frac{312}{49} & & \\ & V = \left\{ \frac{312}{49} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x-\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 35x-112 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-35x} & = & \color{red}{35x} -112 \color{blue}{-35x} \color{blue}{-6} \\\Leftrightarrow & -33x& = & -118 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-118}{-33} \\\Leftrightarrow & x = \frac{118}{33} & & \\ & V = \left\{ \frac{118}{33} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & -28x+70 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{+28x} & = & \color{red}{-28x} +70 \color{blue}{+28x} \color{blue}{-20} \\\Leftrightarrow & 63x& = & 50 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{50}{63} \\\Leftrightarrow & x = \frac{50}{63} & & \\ & V = \left\{ \frac{50}{63} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{4}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & 312x+3120 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{-312x} & = & \color{red}{312x} +3120 \color{blue}{-312x} \color{blue}{+90} \\\Leftrightarrow & -247x& = & 3210 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{3210}{-247} \\\Leftrightarrow & x = \frac{-3210}{247} & & \\ & V = \left\{ \frac{-3210}{247} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & 20x-240 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{-20x} & = & \color{red}{20x} -240 \color{blue}{-20x} \color{blue}{-18} \\\Leftrightarrow & -5x& = & -258 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-258}{-5} \\\Leftrightarrow & x = \frac{258}{5} & & \\ & V = \left\{ \frac{258}{5} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{-7}{4}x+1 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x+\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+40& = & -245x+140 \\\Leftrightarrow & 28x \color{red}{+40} \color{blue}{-40} \color{blue}{+245x} & = & \color{red}{-245x} +140 \color{blue}{+245x} \color{blue}{-40} \\\Leftrightarrow & 273x& = & 100 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{100}{273} \\\Leftrightarrow & x = \frac{100}{273} & & \\ & V = \left\{ \frac{100}{273} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{63}{ \color{blue}{105} }x-\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+60& = & 63x-630 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-63x} & = & \color{red}{63x} -630 \color{blue}{-63x} \color{blue}{-60} \\\Leftrightarrow & -28x& = & -690 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{-690}{-28} \\\Leftrightarrow & x = \frac{345}{14} & & \\ & V = \left\{ \frac{345}{14} \right\} & \\\end{align}\)