Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{12}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{84}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x-25& = & 84x+360 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-84x} & = & \color{red}{84x} +360 \color{blue}{-84x} \color{blue}{+25} \\\Leftrightarrow & -74x& = & 385 \\\Leftrightarrow & \frac{-74x}{ \color{red}{-74} }& = & \frac{385}{-74} \\\Leftrightarrow & x = \frac{-385}{74} & & \\ & V = \left\{ \frac{-385}{74} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{13}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 30 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+30& = & -130x+78 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{+130x} & = & \color{red}{-130x} +78 \color{blue}{+130x} \color{blue}{-30} \\\Leftrightarrow & 169x& = & 48 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{48}{169} \\\Leftrightarrow & x = \frac{48}{169} & & \\ & V = \left\{ \frac{48}{169} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x+\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+14& = & -84x+630 \\\Leftrightarrow & 15x \color{red}{+14} \color{blue}{-14} \color{blue}{+84x} & = & \color{red}{-84x} +630 \color{blue}{+84x} \color{blue}{-14} \\\Leftrightarrow & 99x& = & 616 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{616}{99} \\\Leftrightarrow & x = \frac{56}{9} & & \\ & V = \left\{ \frac{56}{9} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-12& = & 33x-330 \\\Leftrightarrow & 22x \color{red}{-12} \color{blue}{+12} \color{blue}{-33x} & = & \color{red}{33x} -330 \color{blue}{-33x} \color{blue}{+12} \\\Leftrightarrow & -11x& = & -318 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-318}{-11} \\\Leftrightarrow & x = \frac{318}{11} & & \\ & V = \left\{ \frac{318}{11} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{45}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+8& = & 45x-90 \\\Leftrightarrow & 10x \color{red}{+8} \color{blue}{-8} \color{blue}{-45x} & = & \color{red}{45x} -90 \color{blue}{-45x} \color{blue}{-8} \\\Leftrightarrow & -35x& = & -98 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-98}{-35} \\\Leftrightarrow & x = \frac{14}{5} & & \\ & V = \left\{ \frac{14}{5} \right\} & \\\end{align}\)
6. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{63}{ \color{blue}{105} }x-\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+30& = & 63x-630 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-63x} & = & \color{red}{63x} -630 \color{blue}{-63x} \color{blue}{-30} \\\Leftrightarrow & -28x& = & -660 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{-660}{-28} \\\Leftrightarrow & x = \frac{165}{7} & & \\ & V = \left\{ \frac{165}{7} \right\} & \\\end{align}\)
7. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{3}{4}x+2 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }+ \frac{ 56 }{ \color{blue}{252} })& = & (\frac{189}{ \color{blue}{252} }x+\frac{504}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x+56& = & 189x+504 \\\Leftrightarrow & 36x \color{red}{+56} \color{blue}{-56} \color{blue}{-189x} & = & \color{red}{189x} +504 \color{blue}{-189x} \color{blue}{-56} \\\Leftrightarrow & -153x& = & 448 \\\Leftrightarrow & \frac{-153x}{ \color{red}{-153} }& = & \frac{448}{-153} \\\Leftrightarrow & x = \frac{-448}{153} & & \\ & V = \left\{ \frac{-448}{153} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -24x-60 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} -60 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 29x& = & -85 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-85}{29} \\\Leftrightarrow & x = \frac{-85}{29} & & \\ & V = \left\{ \frac{-85}{29} \right\} & \\\end{align}\)
9. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{11}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 30 }{ \color{blue}{110} })& = & (\frac{-44}{ \color{blue}{110} }x+\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-30& = & -44x+880 \\\Leftrightarrow & 55x \color{red}{-30} \color{blue}{+30} \color{blue}{+44x} & = & \color{red}{-44x} +880 \color{blue}{+44x} \color{blue}{+30} \\\Leftrightarrow & 99x& = & 910 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{910}{99} \\\Leftrightarrow & x = \frac{910}{99} & & \\ & V = \left\{ \frac{910}{99} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -24x+150 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+24x} & = & \color{red}{-24x} +150 \color{blue}{+24x} \color{blue}{-9} \\\Leftrightarrow & 29x& = & 141 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{141}{29} \\\Leftrightarrow & x = \frac{141}{29} & & \\ & V = \left\{ \frac{141}{29} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{98}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & 98x+70 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{-98x} & = & \color{red}{98x} +70 \color{blue}{-98x} \color{blue}{-20} \\\Leftrightarrow & -63x& = & 50 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{50}{-63} \\\Leftrightarrow & x = \frac{-50}{63} & & \\ & V = \left\{ \frac{-50}{63} \right\} & \\\end{align}\)
12. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{275}{ \color{blue}{165} }x+\frac{165}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+75& = & 275x+165 \\\Leftrightarrow & 33x \color{red}{+75} \color{blue}{-75} \color{blue}{-275x} & = & \color{red}{275x} +165 \color{blue}{-275x} \color{blue}{-75} \\\Leftrightarrow & -242x& = & 90 \\\Leftrightarrow & \frac{-242x}{ \color{red}{-242} }& = & \frac{90}{-242} \\\Leftrightarrow & x = \frac{-45}{121} & & \\ & V = \left\{ \frac{-45}{121} \right\} & \\\end{align}\)