Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x+\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+16& = & -24x+288 \\\Leftrightarrow & 9x \color{red}{+16} \color{blue}{-16} \color{blue}{+24x} & = & \color{red}{-24x} +288 \color{blue}{+24x} \color{blue}{-16} \\\Leftrightarrow & 33x& = & 272 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{272}{33} \\\Leftrightarrow & x = \frac{272}{33} & & \\ & V = \left\{ \frac{272}{33} \right\} & \\\end{align}\)
2. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 63 }{ \color{blue}{273} })& = & (\frac{-182}{ \color{blue}{273} }x+\frac{2184}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+63& = & -182x+2184 \\\Leftrightarrow & 39x \color{red}{+63} \color{blue}{-63} \color{blue}{+182x} & = & \color{red}{-182x} +2184 \color{blue}{+182x} \color{blue}{-63} \\\Leftrightarrow & 221x& = & 2121 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{2121}{221} \\\Leftrightarrow & x = \frac{2121}{221} & & \\ & V = \left\{ \frac{2121}{221} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{1}{4}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+48& = & 39x-780 \\\Leftrightarrow & 52x \color{red}{+48} \color{blue}{-48} \color{blue}{-39x} & = & \color{red}{39x} -780 \color{blue}{-39x} \color{blue}{-48} \\\Leftrightarrow & 13x& = & -828 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-828}{13} \\\Leftrightarrow & x = \frac{-828}{13} & & \\ & V = \left\{ \frac{-828}{13} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{99}{ \color{blue}{66} }x-\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-30& = & 99x-198 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-99x} & = & \color{red}{99x} -198 \color{blue}{-99x} \color{blue}{+30} \\\Leftrightarrow & -77x& = & -168 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-168}{-77} \\\Leftrightarrow & x = \frac{24}{11} & & \\ & V = \left\{ \frac{24}{11} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & 66x-1650 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{-66x} & = & \color{red}{66x} -1650 \color{blue}{-66x} \color{blue}{-90} \\\Leftrightarrow & -11x& = & -1740 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-1740}{-11} \\\Leftrightarrow & x = \frac{1740}{11} & & \\ & V = \left\{ \frac{1740}{11} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-32}{ \color{blue}{12} }x+\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+10& = & -32x+84 \\\Leftrightarrow & 3x \color{red}{+10} \color{blue}{-10} \color{blue}{+32x} & = & \color{red}{-32x} +84 \color{blue}{+32x} \color{blue}{-10} \\\Leftrightarrow & 35x& = & 74 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{74}{35} \\\Leftrightarrow & x = \frac{74}{35} & & \\ & V = \left\{ \frac{74}{35} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{6}{5}x-6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 168x-840 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-168x} & = & \color{red}{168x} -840 \color{blue}{-168x} \color{blue}{-80} \\\Leftrightarrow & -133x& = & -920 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-920}{-133} \\\Leftrightarrow & x = \frac{920}{133} & & \\ & V = \left\{ \frac{920}{133} \right\} & \\\end{align}\)
8. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x-\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+15& = & -64x-80 \\\Leftrightarrow & 40x \color{red}{+15} \color{blue}{-15} \color{blue}{+64x} & = & \color{red}{-64x} -80 \color{blue}{+64x} \color{blue}{-15} \\\Leftrightarrow & 104x& = & -95 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{-95}{104} \\\Leftrightarrow & x = \frac{-95}{104} & & \\ & V = \left\{ \frac{-95}{104} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{-3}{4}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+40& = & -105x-560 \\\Leftrightarrow & 28x \color{red}{+40} \color{blue}{-40} \color{blue}{+105x} & = & \color{red}{-105x} -560 \color{blue}{+105x} \color{blue}{-40} \\\Leftrightarrow & 133x& = & -600 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-600}{133} \\\Leftrightarrow & x = \frac{-600}{133} & & \\ & V = \left\{ \frac{-600}{133} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & 36x-240 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{-36x} & = & \color{red}{36x} -240 \color{blue}{-36x} \color{blue}{-4} \\\Leftrightarrow & -21x& = & -244 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-244}{-21} \\\Leftrightarrow & x = \frac{244}{21} & & \\ & V = \left\{ \frac{244}{21} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{1}{3}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-25& = & 10x-60 \\\Leftrightarrow & 6x \color{red}{-25} \color{blue}{+25} \color{blue}{-10x} & = & \color{red}{10x} -60 \color{blue}{-10x} \color{blue}{+25} \\\Leftrightarrow & -4x& = & -35 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-35}{-4} \\\Leftrightarrow & x = \frac{35}{4} & & \\ & V = \left\{ \frac{35}{4} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & -40x-60 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+40x} & = & \color{red}{-40x} -60 \color{blue}{+40x} \color{blue}{+8} \\\Leftrightarrow & 55x& = & -52 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{-52}{55} \\\Leftrightarrow & x = \frac{-52}{55} & & \\ & V = \left\{ \frac{-52}{55} \right\} & \\\end{align}\)