Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{135}{ \color{blue}{90} }x-\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 135x-180 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-135x} & = & \color{red}{135x} -180 \color{blue}{-135x} \color{blue}{+20} \\\Leftrightarrow & -117x& = & -160 \\\Leftrightarrow & \frac{-117x}{ \color{red}{-117} }& = & \frac{-160}{-117} \\\Leftrightarrow & x = \frac{160}{117} & & \\ & V = \left\{ \frac{160}{117} \right\} & \\\end{align}\)
2. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x+\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & 12x+288 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} +288 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & 296 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{296}{-3} \\\Leftrightarrow & x = \frac{-296}{3} & & \\ & V = \left\{ \frac{-296}{3} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & 35x-1260 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{-35x} & = & \color{red}{35x} -1260 \color{blue}{-35x} \color{blue}{-90} \\\Leftrightarrow & 7x& = & -1350 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1350}{7} \\\Leftrightarrow & x = \frac{-1350}{7} & & \\ & V = \left\{ \frac{-1350}{7} \right\} & \\\end{align}\)
4. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{9}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & 9x+108 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{-9x} & = & \color{red}{9x} +108 \color{blue}{-9x} \color{blue}{-8} \\\Leftrightarrow & 3x& = & 100 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{100}{3} \\\Leftrightarrow & x = \frac{100}{3} & & \\ & V = \left\{ \frac{100}{3} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & -56x-490 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{+56x} & = & \color{red}{-56x} -490 \color{blue}{+56x} \color{blue}{+15} \\\Leftrightarrow & 91x& = & -475 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-475}{91} \\\Leftrightarrow & x = \frac{-475}{91} & & \\ & V = \left\{ \frac{-475}{91} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 22x+396 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-22x} & = & \color{red}{22x} +396 \color{blue}{-22x} \color{blue}{+30} \\\Leftrightarrow & 11x& = & 426 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{426}{11} \\\Leftrightarrow & x = \frac{426}{11} & & \\ & V = \left\{ \frac{426}{11} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & -12x-36 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{+12x} & = & \color{red}{-12x} -36 \color{blue}{+12x} \color{blue}{-4} \\\Leftrightarrow & 21x& = & -40 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-40}{21} \\\Leftrightarrow & x = \frac{-40}{21} & & \\ & V = \left\{ \frac{-40}{21} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{105}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-15& = & 105x-140 \\\Leftrightarrow & 14x \color{red}{-15} \color{blue}{+15} \color{blue}{-105x} & = & \color{red}{105x} -140 \color{blue}{-105x} \color{blue}{+15} \\\Leftrightarrow & -91x& = & -125 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-125}{-91} \\\Leftrightarrow & x = \frac{125}{91} & & \\ & V = \left\{ \frac{125}{91} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -156x+3120 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+156x} & = & \color{red}{-156x} +3120 \color{blue}{+156x} \color{blue}{-90} \\\Leftrightarrow & 221x& = & 3030 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{3030}{221} \\\Leftrightarrow & x = \frac{3030}{221} & & \\ & V = \left\{ \frac{3030}{221} \right\} & \\\end{align}\)
10. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }- \frac{ 4 }{ \color{blue}{15} })& = & (\frac{-12}{ \color{blue}{15} }x-\frac{30}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x-4& = & -12x-30 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+12x} & = & \color{red}{-12x} -30 \color{blue}{+12x} \color{blue}{+4} \\\Leftrightarrow & 17x& = & -26 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-26}{17} \\\Leftrightarrow & x = \frac{-26}{17} & & \\ & V = \left\{ \frac{-26}{17} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-8}{3}x+3 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-48}{ \color{blue}{18} }x+\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & -48x+54 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+48x} & = & \color{red}{-48x} +54 \color{blue}{+48x} \color{blue}{+8} \\\Leftrightarrow & 57x& = & 62 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{62}{57} \\\Leftrightarrow & x = \frac{62}{57} & & \\ & V = \left\{ \frac{62}{57} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-18& = & 21x+336 \\\Leftrightarrow & 28x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} +336 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & 7x& = & 354 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{354}{7} \\\Leftrightarrow & x = \frac{354}{7} & & \\ & V = \left\{ \frac{354}{7} \right\} & \\\end{align}\)