Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}-\frac{3}{16}=\frac{-2}{3}x+6\)
2. \(\frac{x}{2}-\frac{5}{11}=\frac{-2}{3}x-6\)
3. \(\frac{x}{7}-\frac{5}{6}=\frac{1}{2}x+2\)
4. \(\frac{x}{2}-\frac{3}{8}=\frac{7}{3}x+4\)
5. \(\frac{x}{6}+\frac{5}{16}=\frac{1}{5}x+4\)
6. \(\frac{x}{7}+\frac{3}{10}=\frac{4}{3}x+3\)
7. \(\frac{x}{5}-\frac{2}{7}=\frac{-3}{4}x-5\)
8. \(\frac{x}{4}+\frac{2}{15}=\frac{-5}{3}x+7\)
9. \(\frac{x}{6}-\frac{3}{13}=\frac{1}{5}x+5\)
10. \(\frac{x}{7}+\frac{4}{7}=\frac{6}{5}x-8\)
11. \(\frac{x}{6}+\frac{2}{13}=\frac{-4}{5}x+5\)
12. \(\frac{x}{3}+\frac{4}{15}=\frac{6}{5}x+5\)

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

Verbetersleutel

1. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 63 }{ \color{blue}{336} })& = & (\frac{-224}{ \color{blue}{336} }x+\frac{2016}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-63& = & -224x+2016 \\\Leftrightarrow & 48x \color{red}{-63} \color{blue}{+63} \color{blue}{+224x} & = & \color{red}{-224x} +2016 \color{blue}{+224x} \color{blue}{+63} \\\Leftrightarrow & 272x& = & 2079 \\\Leftrightarrow & \frac{272x}{ \color{red}{272} }& = & \frac{2079}{272} \\\Leftrightarrow & x = \frac{2079}{272} & & \\ & V = \left\{ \frac{2079}{272} \right\} & \\\end{align}\)
2. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & -44x-396 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{+44x} & = & \color{red}{-44x} -396 \color{blue}{+44x} \color{blue}{+30} \\\Leftrightarrow & 77x& = & -366 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-366}{77} \\\Leftrightarrow & x = \frac{-366}{77} & & \\ & V = \left\{ \frac{-366}{77} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 21x+84 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{+35} \\\Leftrightarrow & -15x& = & 119 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{119}{-15} \\\Leftrightarrow & x = \frac{-119}{15} & & \\ & V = \left\{ \frac{-119}{15} \right\} & \\\end{align}\)
4. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{8}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{56}{ \color{blue}{24} }x+\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-9& = & 56x+96 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{-56x} & = & \color{red}{56x} +96 \color{blue}{-56x} \color{blue}{+9} \\\Leftrightarrow & -44x& = & 105 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{105}{-44} \\\Leftrightarrow & x = \frac{-105}{44} & & \\ & V = \left\{ \frac{-105}{44} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x+\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & 48x+960 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-48x} & = & \color{red}{48x} +960 \color{blue}{-48x} \color{blue}{-75} \\\Leftrightarrow & -8x& = & 885 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{885}{-8} \\\Leftrightarrow & x = \frac{-885}{8} & & \\ & V = \left\{ \frac{-885}{8} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{4}{3}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 63 }{ \color{blue}{210} })& = & (\frac{280}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+63& = & 280x+630 \\\Leftrightarrow & 30x \color{red}{+63} \color{blue}{-63} \color{blue}{-280x} & = & \color{red}{280x} +630 \color{blue}{-280x} \color{blue}{-63} \\\Leftrightarrow & -250x& = & 567 \\\Leftrightarrow & \frac{-250x}{ \color{red}{-250} }& = & \frac{567}{-250} \\\Leftrightarrow & x = \frac{-567}{250} & & \\ & V = \left\{ \frac{-567}{250} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-40& = & -105x-700 \\\Leftrightarrow & 28x \color{red}{-40} \color{blue}{+40} \color{blue}{+105x} & = & \color{red}{-105x} -700 \color{blue}{+105x} \color{blue}{+40} \\\Leftrightarrow & 133x& = & -660 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-660}{133} \\\Leftrightarrow & x = \frac{-660}{133} & & \\ & V = \left\{ \frac{-660}{133} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -100x+420 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+100x} & = & \color{red}{-100x} +420 \color{blue}{+100x} \color{blue}{-8} \\\Leftrightarrow & 115x& = & 412 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{412}{115} \\\Leftrightarrow & x = \frac{412}{115} & & \\ & V = \left\{ \frac{412}{115} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & 78x+1950 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{-78x} & = & \color{red}{78x} +1950 \color{blue}{-78x} \color{blue}{+90} \\\Leftrightarrow & -13x& = & 2040 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{2040}{-13} \\\Leftrightarrow & x = \frac{-2040}{13} & & \\ & V = \left\{ \frac{-2040}{13} \right\} & \\\end{align}\)
10. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{42}{ \color{blue}{35} }x-\frac{280}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 42x-280 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-42x} & = & \color{red}{42x} -280 \color{blue}{-42x} \color{blue}{-20} \\\Leftrightarrow & -37x& = & -300 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{-300}{-37} \\\Leftrightarrow & x = \frac{300}{37} & & \\ & V = \left\{ \frac{300}{37} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & -312x+1950 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+312x} & = & \color{red}{-312x} +1950 \color{blue}{+312x} \color{blue}{-60} \\\Leftrightarrow & 377x& = & 1890 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{1890}{377} \\\Leftrightarrow & x = \frac{1890}{377} & & \\ & V = \left\{ \frac{1890}{377} \right\} & \\\end{align}\)
12. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 4 }{ \color{blue}{15} })& = & (\frac{18}{ \color{blue}{15} }x+\frac{75}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+4& = & 18x+75 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-18x} & = & \color{red}{18x} +75 \color{blue}{-18x} \color{blue}{-4} \\\Leftrightarrow & -13x& = & 71 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{71}{-13} \\\Leftrightarrow & x = \frac{-71}{13} & & \\ & V = \left\{ \frac{-71}{13} \right\} & \\\end{align}\)