Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & -98x+70 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{+98x} & = & \color{red}{-98x} +70 \color{blue}{+98x} \color{blue}{+15} \\\Leftrightarrow & 133x& = & 85 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{85}{133} \\\Leftrightarrow & x = \frac{85}{133} & & \\ & V = \left\{ \frac{85}{133} \right\} & \\\end{align}\)
2. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 42 }{ \color{blue}{231} })& = & (\frac{539}{ \color{blue}{231} }x-\frac{1617}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+42& = & 539x-1617 \\\Leftrightarrow & 33x \color{red}{+42} \color{blue}{-42} \color{blue}{-539x} & = & \color{red}{539x} -1617 \color{blue}{-539x} \color{blue}{-42} \\\Leftrightarrow & -506x& = & -1659 \\\Leftrightarrow & \frac{-506x}{ \color{red}{-506} }& = & \frac{-1659}{-506} \\\Leftrightarrow & x = \frac{1659}{506} & & \\ & V = \left\{ \frac{1659}{506} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x-\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-15& = & 16x-160 \\\Leftrightarrow & 40x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} -160 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & 24x& = & -145 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{-145}{24} \\\Leftrightarrow & x = \frac{-145}{24} & & \\ & V = \left\{ \frac{-145}{24} \right\} & \\\end{align}\)
4. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x-\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & 12x-216 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-12x} & = & \color{red}{12x} -216 \color{blue}{-12x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & -224 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-224}{-3} \\\Leftrightarrow & x = \frac{224}{3} & & \\ & V = \left\{ \frac{224}{3} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 15x+30 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-15x} & = & \color{red}{15x} +30 \color{blue}{-15x} \color{blue}{-8} \\\Leftrightarrow & -9x& = & 22 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{22}{-9} \\\Leftrightarrow & x = \frac{-22}{9} & & \\ & V = \left\{ \frac{-22}{9} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 50 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+50& = & 35x+350 \\\Leftrightarrow & 14x \color{red}{+50} \color{blue}{-50} \color{blue}{-35x} & = & \color{red}{35x} +350 \color{blue}{-35x} \color{blue}{-50} \\\Leftrightarrow & -21x& = & 300 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{300}{-21} \\\Leftrightarrow & x = \frac{-100}{7} & & \\ & V = \left\{ \frac{-100}{7} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 36x+180 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-36x} & = & \color{red}{36x} +180 \color{blue}{-36x} \color{blue}{-9} \\\Leftrightarrow & -26x& = & 171 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{171}{-26} \\\Leftrightarrow & x = \frac{-171}{26} & & \\ & V = \left\{ \frac{-171}{26} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{3}{4}x+7 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{99}{ \color{blue}{132} }x+\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+60& = & 99x+924 \\\Leftrightarrow & 44x \color{red}{+60} \color{blue}{-60} \color{blue}{-99x} & = & \color{red}{99x} +924 \color{blue}{-99x} \color{blue}{-60} \\\Leftrightarrow & -55x& = & 864 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{864}{-55} \\\Leftrightarrow & x = \frac{-864}{55} & & \\ & V = \left\{ \frac{-864}{55} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 98x+42 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-98x} & = & \color{red}{98x} +42 \color{blue}{-98x} \color{blue}{+12} \\\Leftrightarrow & -77x& = & 54 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{54}{-77} \\\Leftrightarrow & x = \frac{-54}{77} & & \\ & V = \left\{ \frac{-54}{77} \right\} & \\\end{align}\)
10. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x+70 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} +70 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & 76 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{76}{-5} \\\Leftrightarrow & x = \frac{-76}{5} & & \\ & V = \left\{ \frac{-76}{5} \right\} & \\\end{align}\)
11. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{3}{4}x+6 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{60}{ \color{blue}{80} }x+\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 60x+480 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-60x} & = & \color{red}{60x} +480 \color{blue}{-60x} \color{blue}{-15} \\\Leftrightarrow & -44x& = & 465 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{465}{-44} \\\Leftrightarrow & x = \frac{-465}{44} & & \\ & V = \left\{ \frac{-465}{44} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{2}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{28}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 28x-42 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-28x} & = & \color{red}{28x} -42 \color{blue}{-28x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -66 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-66}{-7} \\\Leftrightarrow & x = \frac{66}{7} & & \\ & V = \left\{ \frac{66}{7} \right\} & \\\end{align}\)