Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{2}{9}=\frac{1}{2}x+6\)
2. \(\frac{x}{2}-\frac{3}{14}=\frac{7}{3}x+5\)
3. \(\frac{x}{5}+\frac{2}{15}=\frac{-7}{4}x+1\)
4. \(\frac{x}{7}+\frac{3}{10}=\frac{-5}{6}x-7\)
5. \(\frac{x}{3}-\frac{2}{15}=\frac{5}{4}x-7\)
6. \(\frac{x}{6}+\frac{2}{15}=\frac{6}{5}x+7\)
7. \(\frac{x}{3}+\frac{5}{9}=\frac{5}{4}x+6\)
8. \(\frac{x}{7}-\frac{5}{7}=\frac{-5}{6}x-3\)
9. \(\frac{x}{4}+\frac{4}{13}=\frac{-5}{3}x-1\)
10. \(\frac{x}{5}+\frac{5}{7}=\frac{1}{6}x+1\)
11. \(\frac{x}{3}-\frac{3}{8}=\frac{7}{5}x+7\)
12. \(\frac{x}{4}+\frac{3}{14}=\frac{6}{5}x+5\)

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

Verbetersleutel

1. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{756}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & 63x+756 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{-63x} & = & \color{red}{63x} +756 \color{blue}{-63x} \color{blue}{-28} \\\Leftrightarrow & -45x& = & 728 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{728}{-45} \\\Leftrightarrow & x = \frac{-728}{45} & & \\ & V = \left\{ \frac{-728}{45} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & 98x+210 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{-98x} & = & \color{red}{98x} +210 \color{blue}{-98x} \color{blue}{+9} \\\Leftrightarrow & -77x& = & 219 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{219}{-77} \\\Leftrightarrow & x = \frac{-219}{77} & & \\ & V = \left\{ \frac{-219}{77} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{-7}{4}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-105}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+8& = & -105x+60 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{+105x} & = & \color{red}{-105x} +60 \color{blue}{+105x} \color{blue}{-8} \\\Leftrightarrow & 117x& = & 52 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{52}{117} \\\Leftrightarrow & x = \frac{4}{9} & & \\ & V = \left\{ \frac{4}{9} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 63 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+63& = & -175x-1470 \\\Leftrightarrow & 30x \color{red}{+63} \color{blue}{-63} \color{blue}{+175x} & = & \color{red}{-175x} -1470 \color{blue}{+175x} \color{blue}{-63} \\\Leftrightarrow & 205x& = & -1533 \\\Leftrightarrow & \frac{205x}{ \color{red}{205} }& = & \frac{-1533}{205} \\\Leftrightarrow & x = \frac{-1533}{205} & & \\ & V = \left\{ \frac{-1533}{205} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{5}{4}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-8& = & 75x-420 \\\Leftrightarrow & 20x \color{red}{-8} \color{blue}{+8} \color{blue}{-75x} & = & \color{red}{75x} -420 \color{blue}{-75x} \color{blue}{+8} \\\Leftrightarrow & -55x& = & -412 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-412}{-55} \\\Leftrightarrow & x = \frac{412}{55} & & \\ & V = \left\{ \frac{412}{55} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 36x+210 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-36x} & = & \color{red}{36x} +210 \color{blue}{-36x} \color{blue}{-4} \\\Leftrightarrow & -31x& = & 206 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{206}{-31} \\\Leftrightarrow & x = \frac{-206}{31} & & \\ & V = \left\{ \frac{-206}{31} \right\} & \\\end{align}\)
7. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{9}& = & \frac{5}{4}x+6 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 20 }{ \color{blue}{36} })& = & (\frac{45}{ \color{blue}{36} }x+\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+20& = & 45x+216 \\\Leftrightarrow & 12x \color{red}{+20} \color{blue}{-20} \color{blue}{-45x} & = & \color{red}{45x} +216 \color{blue}{-45x} \color{blue}{-20} \\\Leftrightarrow & -33x& = & 196 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{196}{-33} \\\Leftrightarrow & x = \frac{-196}{33} & & \\ & V = \left\{ \frac{-196}{33} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{-5}{6}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-30& = & -35x-126 \\\Leftrightarrow & 6x \color{red}{-30} \color{blue}{+30} \color{blue}{+35x} & = & \color{red}{-35x} -126 \color{blue}{+35x} \color{blue}{+30} \\\Leftrightarrow & 41x& = & -96 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{-96}{41} \\\Leftrightarrow & x = \frac{-96}{41} & & \\ & V = \left\{ \frac{-96}{41} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & -260x-156 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{+260x} & = & \color{red}{-260x} -156 \color{blue}{+260x} \color{blue}{-48} \\\Leftrightarrow & 299x& = & -204 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{-204}{299} \\\Leftrightarrow & x = \frac{-204}{299} & & \\ & V = \left\{ \frac{-204}{299} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{7}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+150& = & 35x+210 \\\Leftrightarrow & 42x \color{red}{+150} \color{blue}{-150} \color{blue}{-35x} & = & \color{red}{35x} +210 \color{blue}{-35x} \color{blue}{-150} \\\Leftrightarrow & 7x& = & 60 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{60}{7} \\\Leftrightarrow & x = \frac{60}{7} & & \\ & V = \left\{ \frac{60}{7} \right\} & \\\end{align}\)
11. \(\text{120 is het kleinste gemene veelvoud van 3, 8 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{120.} (\frac{40x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{168}{ \color{blue}{120} }x+\frac{840}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 40x-45& = & 168x+840 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-168x} & = & \color{red}{168x} +840 \color{blue}{-168x} \color{blue}{+45} \\\Leftrightarrow & -128x& = & 885 \\\Leftrightarrow & \frac{-128x}{ \color{red}{-128} }& = & \frac{885}{-128} \\\Leftrightarrow & x = \frac{-885}{128} & & \\ & V = \left\{ \frac{-885}{128} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 30 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+30& = & 168x+700 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-168x} & = & \color{red}{168x} +700 \color{blue}{-168x} \color{blue}{-30} \\\Leftrightarrow & -133x& = & 670 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{670}{-133} \\\Leftrightarrow & x = \frac{-670}{133} & & \\ & V = \left\{ \frac{-670}{133} \right\} & \\\end{align}\)