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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x+\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 48x+240 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-48x} & = & \color{red}{48x} +240 \color{blue}{-48x} \color{blue}{+45} \\\Leftrightarrow & -8x& = & 285 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{285}{-8} \\\Leftrightarrow & x = \frac{-285}{8} & & \\ & V = \left\{ \frac{-285}{8} \right\} & \\\end{align}\)
2. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{3}{5}x-3 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 2 }{ \color{blue}{15} })& = & (\frac{9}{ \color{blue}{15} }x-\frac{45}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+2& = & 9x-45 \\\Leftrightarrow & 5x \color{red}{+2} \color{blue}{-2} \color{blue}{-9x} & = & \color{red}{9x} -45 \color{blue}{-9x} \color{blue}{-2} \\\Leftrightarrow & -4x& = & -47 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-47}{-4} \\\Leftrightarrow & x = \frac{47}{4} & & \\ & V = \left\{ \frac{47}{4} \right\} & \\\end{align}\)
3. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{8}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 35 }{ \color{blue}{56} })& = & (\frac{28}{ \color{blue}{56} }x-\frac{168}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+35& = & 28x-168 \\\Leftrightarrow & 8x \color{red}{+35} \color{blue}{-35} \color{blue}{-28x} & = & \color{red}{28x} -168 \color{blue}{-28x} \color{blue}{-35} \\\Leftrightarrow & -20x& = & -203 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{-203}{-20} \\\Leftrightarrow & x = \frac{203}{20} & & \\ & V = \left\{ \frac{203}{20} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 100 }{ \color{blue}{220} })& = & (\frac{264}{ \color{blue}{220} }x-\frac{220}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-100& = & 264x-220 \\\Leftrightarrow & 55x \color{red}{-100} \color{blue}{+100} \color{blue}{-264x} & = & \color{red}{264x} -220 \color{blue}{-264x} \color{blue}{+100} \\\Leftrightarrow & -209x& = & -120 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-120}{-209} \\\Leftrightarrow & x = \frac{120}{209} & & \\ & V = \left\{ \frac{120}{209} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 7, 14 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-15& = & 84x+70 \\\Leftrightarrow & 10x \color{red}{-15} \color{blue}{+15} \color{blue}{-84x} & = & \color{red}{84x} +70 \color{blue}{-84x} \color{blue}{+15} \\\Leftrightarrow & -74x& = & 85 \\\Leftrightarrow & \frac{-74x}{ \color{red}{-74} }& = & \frac{85}{-74} \\\Leftrightarrow & x = \frac{-85}{74} & & \\ & V = \left\{ \frac{-85}{74} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{7}{4}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{147}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 147x+336 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-147x} & = & \color{red}{147x} +336 \color{blue}{-147x} \color{blue}{+36} \\\Leftrightarrow & -119x& = & 372 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{372}{-119} \\\Leftrightarrow & x = \frac{-372}{119} & & \\ & V = \left\{ \frac{-372}{119} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 42x-1470 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-42x} & = & \color{red}{42x} -1470 \color{blue}{-42x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & -1410 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-1410}{-7} \\\Leftrightarrow & x = \frac{1410}{7} & & \\ & V = \left\{ \frac{1410}{7} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & -176x+66 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{+176x} & = & \color{red}{-176x} +66 \color{blue}{+176x} \color{blue}{+12} \\\Leftrightarrow & 209x& = & 78 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{78}{209} \\\Leftrightarrow & x = \frac{78}{209} & & \\ & V = \left\{ \frac{78}{209} \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-63}{ \color{blue}{36} }x+\frac{252}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & -63x+252 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{+63x} & = & \color{red}{-63x} +252 \color{blue}{+63x} \color{blue}{-8} \\\Leftrightarrow & 75x& = & 244 \\\Leftrightarrow & \frac{75x}{ \color{red}{75} }& = & \frac{244}{75} \\\Leftrightarrow & x = \frac{244}{75} & & \\ & V = \left\{ \frac{244}{75} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{2}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 12x-180 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-12x} & = & \color{red}{12x} -180 \color{blue}{-12x} \color{blue}{+4} \\\Leftrightarrow & -7x& = & -176 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-176}{-7} \\\Leftrightarrow & x = \frac{176}{7} & & \\ & V = \left\{ \frac{176}{7} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 98x-210 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-98x} & = & \color{red}{98x} -210 \color{blue}{-98x} \color{blue}{+18} \\\Leftrightarrow & -77x& = & -192 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-192}{-77} \\\Leftrightarrow & x = \frac{192}{77} & & \\ & V = \left\{ \frac{192}{77} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 252x+420 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-252x} & = & \color{red}{252x} +420 \color{blue}{-252x} \color{blue}{+120} \\\Leftrightarrow & -217x& = & 540 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{540}{-217} \\\Leftrightarrow & x = \frac{-540}{217} & & \\ & V = \left\{ \frac{-540}{217} \right\} & \\\end{align}\)