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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & 20x-60 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{-20x} & = & \color{red}{20x} -60 \color{blue}{-20x} \color{blue}{-18} \\\Leftrightarrow & -5x& = & -78 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-78}{-5} \\\Leftrightarrow & x = \frac{78}{5} & & \\ & V = \left\{ \frac{78}{5} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-2}{5}x+6 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x+\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -96x+1440 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+96x} & = & \color{red}{-96x} +1440 \color{blue}{+96x} \color{blue}{-75} \\\Leftrightarrow & 136x& = & 1365 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{1365}{136} \\\Leftrightarrow & x = \frac{1365}{136} & & \\ & V = \left\{ \frac{1365}{136} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{-7}{4}x-7 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-63}{ \color{blue}{36} }x-\frac{252}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & -63x-252 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{+63x} & = & \color{red}{-63x} -252 \color{blue}{+63x} \color{blue}{+8} \\\Leftrightarrow & 75x& = & -244 \\\Leftrightarrow & \frac{75x}{ \color{red}{75} }& = & \frac{-244}{75} \\\Leftrightarrow & x = \frac{-244}{75} & & \\ & V = \left\{ \frac{-244}{75} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 9x-36 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-9x} & = & \color{red}{9x} -36 \color{blue}{-9x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & -28 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-28}{-3} \\\Leftrightarrow & x = \frac{28}{3} & & \\ & V = \left\{ \frac{28}{3} \right\} & \\\end{align}\)
5. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 63 }{ \color{blue}{231} })& = & (\frac{154}{ \color{blue}{231} }x+\frac{462}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-63& = & 154x+462 \\\Leftrightarrow & 33x \color{red}{-63} \color{blue}{+63} \color{blue}{-154x} & = & \color{red}{154x} +462 \color{blue}{-154x} \color{blue}{+63} \\\Leftrightarrow & -121x& = & 525 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{525}{-121} \\\Leftrightarrow & x = \frac{-525}{121} & & \\ & V = \left\{ \frac{-525}{121} \right\} & \\\end{align}\)
6. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & -80x-96 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{+80x} & = & \color{red}{-80x} -96 \color{blue}{+80x} \color{blue}{+9} \\\Leftrightarrow & 104x& = & -87 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{-87}{104} \\\Leftrightarrow & x = \frac{-87}{104} & & \\ & V = \left\{ \frac{-87}{104} \right\} & \\\end{align}\)
7. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{616}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 77x-616 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-77x} & = & \color{red}{77x} -616 \color{blue}{-77x} \color{blue}{-28} \\\Leftrightarrow & -55x& = & -644 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-644}{-55} \\\Leftrightarrow & x = \frac{644}{55} & & \\ & V = \left\{ \frac{644}{55} \right\} & \\\end{align}\)
8. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{7}{5}x-6 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 60 }{ \color{blue}{260} })& = & (\frac{364}{ \color{blue}{260} }x-\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+60& = & 364x-1560 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-364x} & = & \color{red}{364x} -1560 \color{blue}{-364x} \color{blue}{-60} \\\Leftrightarrow & -299x& = & -1620 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{-1620}{-299} \\\Leftrightarrow & x = \frac{1620}{299} & & \\ & V = \left\{ \frac{1620}{299} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & -20x+150 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{+20x} & = & \color{red}{-20x} +150 \color{blue}{+20x} \color{blue}{-4} \\\Leftrightarrow & 35x& = & 146 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{146}{35} \\\Leftrightarrow & x = \frac{146}{35} & & \\ & V = \left\{ \frac{146}{35} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 66x-1980 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-66x} & = & \color{red}{66x} -1980 \color{blue}{-66x} \color{blue}{+150} \\\Leftrightarrow & -11x& = & -1830 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-1830}{-11} \\\Leftrightarrow & x = \frac{1830}{11} & & \\ & V = \left\{ \frac{1830}{11} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+24& = & -44x-264 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+44x} & = & \color{red}{-44x} -264 \color{blue}{+44x} \color{blue}{-24} \\\Leftrightarrow & 77x& = & -288 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-288}{77} \\\Leftrightarrow & x = \frac{-288}{77} & & \\ & V = \left\{ \frac{-288}{77} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -140x+336 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+140x} & = & \color{red}{-140x} +336 \color{blue}{+140x} \color{blue}{-48} \\\Leftrightarrow & 161x& = & 288 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{288}{161} \\\Leftrightarrow & x = \frac{288}{161} & & \\ & V = \left\{ \frac{288}{161} \right\} & \\\end{align}\)