Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{6}-\frac{3}{7}=\frac{2}{5}x+3\)
2. \(\frac{x}{7}+\frac{3}{14}=\frac{1}{6}x-8\)
3. \(\frac{x}{7}-\frac{3}{13}=\frac{-5}{3}x+6\)
4. \(\frac{x}{5}-\frac{2}{9}=\frac{1}{2}x-7\)
5. \(\frac{x}{3}-\frac{2}{7}=\frac{5}{4}x+4\)
6. \(\frac{x}{5}+\frac{4}{7}=\frac{1}{2}x-3\)
7. \(\frac{x}{5}+\frac{3}{7}=\frac{4}{3}x-2\)
8. \(\frac{x}{2}+\frac{2}{7}=\frac{-4}{5}x+8\)
9. \(\frac{x}{7}+\frac{3}{16}=\frac{5}{6}x-3\)
10. \(\frac{x}{7}+\frac{2}{9}=\frac{1}{2}x+1\)
11. \(\frac{x}{5}-\frac{5}{16}=\frac{1}{6}x-2\)
12. \(\frac{x}{3}-\frac{3}{10}=\frac{-4}{5}x+2\)

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{2}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 84x+630 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-84x} & = & \color{red}{84x} +630 \color{blue}{-84x} \color{blue}{+90} \\\Leftrightarrow & -49x& = & 720 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{720}{-49} \\\Leftrightarrow & x = \frac{-720}{49} & & \\ & V = \left\{ \frac{-720}{49} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+9& = & 7x-336 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{-7x} & = & \color{red}{7x} -336 \color{blue}{-7x} \color{blue}{-9} \\\Leftrightarrow & -x& = & -345 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-345}{-1} \\\Leftrightarrow & x = 345 & & \\ & V = \left\{ 345 \right\} & \\\end{align}\)
3. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 63 }{ \color{blue}{273} })& = & (\frac{-455}{ \color{blue}{273} }x+\frac{1638}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-63& = & -455x+1638 \\\Leftrightarrow & 39x \color{red}{-63} \color{blue}{+63} \color{blue}{+455x} & = & \color{red}{-455x} +1638 \color{blue}{+455x} \color{blue}{+63} \\\Leftrightarrow & 494x& = & 1701 \\\Leftrightarrow & \frac{494x}{ \color{red}{494} }& = & \frac{1701}{494} \\\Leftrightarrow & x = \frac{1701}{494} & & \\ & V = \left\{ \frac{1701}{494} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 45x-630 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-45x} & = & \color{red}{45x} -630 \color{blue}{-45x} \color{blue}{+20} \\\Leftrightarrow & -27x& = & -610 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-610}{-27} \\\Leftrightarrow & x = \frac{610}{27} & & \\ & V = \left\{ \frac{610}{27} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{5}{4}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & 105x+336 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{-105x} & = & \color{red}{105x} +336 \color{blue}{-105x} \color{blue}{+24} \\\Leftrightarrow & -77x& = & 360 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{360}{-77} \\\Leftrightarrow & x = \frac{-360}{77} & & \\ & V = \left\{ \frac{-360}{77} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+40& = & 35x-210 \\\Leftrightarrow & 14x \color{red}{+40} \color{blue}{-40} \color{blue}{-35x} & = & \color{red}{35x} -210 \color{blue}{-35x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & -250 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-250}{-21} \\\Leftrightarrow & x = \frac{250}{21} & & \\ & V = \left\{ \frac{250}{21} \right\} & \\\end{align}\)
7. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 45 }{ \color{blue}{105} })& = & (\frac{140}{ \color{blue}{105} }x-\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+45& = & 140x-210 \\\Leftrightarrow & 21x \color{red}{+45} \color{blue}{-45} \color{blue}{-140x} & = & \color{red}{140x} -210 \color{blue}{-140x} \color{blue}{-45} \\\Leftrightarrow & -119x& = & -255 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-255}{-119} \\\Leftrightarrow & x = \frac{15}{7} & & \\ & V = \left\{ \frac{15}{7} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & -56x+560 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{+56x} & = & \color{red}{-56x} +560 \color{blue}{+56x} \color{blue}{-20} \\\Leftrightarrow & 91x& = & 540 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{540}{91} \\\Leftrightarrow & x = \frac{540}{91} & & \\ & V = \left\{ \frac{540}{91} \right\} & \\\end{align}\)
9. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{5}{6}x-3 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 63 }{ \color{blue}{336} })& = & (\frac{280}{ \color{blue}{336} }x-\frac{1008}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+63& = & 280x-1008 \\\Leftrightarrow & 48x \color{red}{+63} \color{blue}{-63} \color{blue}{-280x} & = & \color{red}{280x} -1008 \color{blue}{-280x} \color{blue}{-63} \\\Leftrightarrow & -232x& = & -1071 \\\Leftrightarrow & \frac{-232x}{ \color{red}{-232} }& = & \frac{-1071}{-232} \\\Leftrightarrow & x = \frac{1071}{232} & & \\ & V = \left\{ \frac{1071}{232} \right\} & \\\end{align}\)
10. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{126}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & 63x+126 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{-63x} & = & \color{red}{63x} +126 \color{blue}{-63x} \color{blue}{-28} \\\Leftrightarrow & -45x& = & 98 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{98}{-45} \\\Leftrightarrow & x = \frac{-98}{45} & & \\ & V = \left\{ \frac{-98}{45} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{1}{6}x-2 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x-\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & 40x-480 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{-40x} & = & \color{red}{40x} -480 \color{blue}{-40x} \color{blue}{+75} \\\Leftrightarrow & 8x& = & -405 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-405}{8} \\\Leftrightarrow & x = \frac{-405}{8} & & \\ & V = \left\{ \frac{-405}{8} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & -24x+60 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{+24x} & = & \color{red}{-24x} +60 \color{blue}{+24x} \color{blue}{+9} \\\Leftrightarrow & 34x& = & 69 \\\Leftrightarrow & \frac{34x}{ \color{red}{34} }& = & \frac{69}{34} \\\Leftrightarrow & x = \frac{69}{34} & & \\ & V = \left\{ \frac{69}{34} \right\} & \\\end{align}\)