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. \(\frac{x}{6}-\frac{4}{9}=\frac{1}{5}x+8\)
2. \(\frac{x}{6}-\frac{5}{14}=\frac{-2}{5}x-8\)
3. \(\frac{x}{5}+\frac{4}{9}=\frac{1}{2}x-8\)
4. \(\frac{x}{6}+\frac{3}{16}=\frac{6}{5}x-7\)
5. \(\frac{x}{4}+\frac{4}{9}=\frac{-5}{3}x-6\)
6. \(\frac{x}{7}+\frac{3}{13}=\frac{-5}{3}x+4\)
7. \(\frac{x}{2}-\frac{3}{16}=\frac{5}{3}x+7\)
8. \(\frac{x}{7}+\frac{5}{8}=\frac{1}{2}x-1\)
9. \(\frac{x}{6}+\frac{2}{11}=\frac{1}{5}x-8\)
10. \(\frac{x}{2}+\frac{5}{12}=\frac{2}{3}x-4\)
11. \(\frac{x}{5}+\frac{5}{8}=\frac{1}{2}x-5\)
12. \(\frac{x}{2}-\frac{2}{11}=\frac{1}{3}x+4\)

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & 18x+720 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{-18x} & = & \color{red}{18x} +720 \color{blue}{-18x} \color{blue}{+40} \\\Leftrightarrow & -3x& = & 760 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{760}{-3} \\\Leftrightarrow & x = \frac{-760}{3} & & \\ & V = \left\{ \frac{-760}{3} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{14}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-75& = & -84x-1680 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{+84x} & = & \color{red}{-84x} -1680 \color{blue}{+84x} \color{blue}{+75} \\\Leftrightarrow & 119x& = & -1605 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-1605}{119} \\\Leftrightarrow & x = \frac{-1605}{119} & & \\ & V = \left\{ \frac{-1605}{119} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 45x-720 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-45x} & = & \color{red}{45x} -720 \color{blue}{-45x} \color{blue}{-40} \\\Leftrightarrow & -27x& = & -760 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-760}{-27} \\\Leftrightarrow & x = \frac{760}{27} & & \\ & V = \left\{ \frac{760}{27} \right\} & \\\end{align}\)
4. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x-\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & 288x-1680 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-288x} & = & \color{red}{288x} -1680 \color{blue}{-288x} \color{blue}{-45} \\\Leftrightarrow & -248x& = & -1725 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{-1725}{-248} \\\Leftrightarrow & x = \frac{1725}{248} & & \\ & V = \left\{ \frac{1725}{248} \right\} & \\\end{align}\)
5. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-60}{ \color{blue}{36} }x-\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & -60x-216 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{+60x} & = & \color{red}{-60x} -216 \color{blue}{+60x} \color{blue}{-16} \\\Leftrightarrow & 69x& = & -232 \\\Leftrightarrow & \frac{69x}{ \color{red}{69} }& = & \frac{-232}{69} \\\Leftrightarrow & x = \frac{-232}{69} & & \\ & V = \left\{ \frac{-232}{69} \right\} & \\\end{align}\)
6. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 63 }{ \color{blue}{273} })& = & (\frac{-455}{ \color{blue}{273} }x+\frac{1092}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+63& = & -455x+1092 \\\Leftrightarrow & 39x \color{red}{+63} \color{blue}{-63} \color{blue}{+455x} & = & \color{red}{-455x} +1092 \color{blue}{+455x} \color{blue}{-63} \\\Leftrightarrow & 494x& = & 1029 \\\Leftrightarrow & \frac{494x}{ \color{red}{494} }& = & \frac{1029}{494} \\\Leftrightarrow & x = \frac{1029}{494} & & \\ & V = \left\{ \frac{1029}{494} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{80}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & 80x+336 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{-80x} & = & \color{red}{80x} +336 \color{blue}{-80x} \color{blue}{+9} \\\Leftrightarrow & -56x& = & 345 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{345}{-56} \\\Leftrightarrow & x = \frac{-345}{56} & & \\ & V = \left\{ \frac{-345}{56} \right\} & \\\end{align}\)
8. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{8}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 35 }{ \color{blue}{56} })& = & (\frac{28}{ \color{blue}{56} }x-\frac{56}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+35& = & 28x-56 \\\Leftrightarrow & 8x \color{red}{+35} \color{blue}{-35} \color{blue}{-28x} & = & \color{red}{28x} -56 \color{blue}{-28x} \color{blue}{-35} \\\Leftrightarrow & -20x& = & -91 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{-91}{-20} \\\Leftrightarrow & x = \frac{91}{20} & & \\ & V = \left\{ \frac{91}{20} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{11}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & 66x-2640 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-66x} & = & \color{red}{66x} -2640 \color{blue}{-66x} \color{blue}{-60} \\\Leftrightarrow & -11x& = & -2700 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-2700}{-11} \\\Leftrightarrow & x = \frac{2700}{11} & & \\ & V = \left\{ \frac{2700}{11} \right\} & \\\end{align}\)
10. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{12}& = & \frac{2}{3}x-4 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{8}{ \color{blue}{12} }x-\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x+5& = & 8x-48 \\\Leftrightarrow & 6x \color{red}{+5} \color{blue}{-5} \color{blue}{-8x} & = & \color{red}{8x} -48 \color{blue}{-8x} \color{blue}{-5} \\\Leftrightarrow & -2x& = & -53 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-53}{-2} \\\Leftrightarrow & x = \frac{53}{2} & & \\ & V = \left\{ \frac{53}{2} \right\} & \\\end{align}\)
11. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{20}{ \color{blue}{40} }x-\frac{200}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+25& = & 20x-200 \\\Leftrightarrow & 8x \color{red}{+25} \color{blue}{-25} \color{blue}{-20x} & = & \color{red}{20x} -200 \color{blue}{-20x} \color{blue}{-25} \\\Leftrightarrow & -12x& = & -225 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{-225}{-12} \\\Leftrightarrow & x = \frac{75}{4} & & \\ & V = \left\{ \frac{75}{4} \right\} & \\\end{align}\)
12. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 22x+264 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-22x} & = & \color{red}{22x} +264 \color{blue}{-22x} \color{blue}{+12} \\\Leftrightarrow & 11x& = & 276 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{276}{11} \\\Leftrightarrow & x = \frac{276}{11} & & \\ & V = \left\{ \frac{276}{11} \right\} & \\\end{align}\)