Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-60& = & -105x-280 \\\Leftrightarrow & 28x \color{red}{-60} \color{blue}{+60} \color{blue}{+105x} & = & \color{red}{-105x} -280 \color{blue}{+105x} \color{blue}{+60} \\\Leftrightarrow & 133x& = & -220 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-220}{133} \\\Leftrightarrow & x = \frac{-220}{133} & & \\ & V = \left\{ \frac{-220}{133} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{8}{ \color{blue}{24} }x+\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+15& = & 8x+192 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{-8x} & = & \color{red}{8x} +192 \color{blue}{-8x} \color{blue}{-15} \\\Leftrightarrow & 4x& = & 177 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{177}{4} \\\Leftrightarrow & x = \frac{177}{4} & & \\ & V = \left\{ \frac{177}{4} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & -44x-66 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+44x} & = & \color{red}{-44x} -66 \color{blue}{+44x} \color{blue}{-30} \\\Leftrightarrow & 77x& = & -96 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-96}{77} \\\Leftrightarrow & x = \frac{-96}{77} & & \\ & V = \left\{ \frac{-96}{77} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x+\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & 26x+312 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{-26x} & = & \color{red}{26x} +312 \color{blue}{-26x} \color{blue}{-18} \\\Leftrightarrow & 13x& = & 294 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{294}{13} \\\Leftrightarrow & x = \frac{294}{13} & & \\ & V = \left\{ \frac{294}{13} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 15x-30 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-15x} & = & \color{red}{15x} -30 \color{blue}{-15x} \color{blue}{-4} \\\Leftrightarrow & -9x& = & -34 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-34}{-9} \\\Leftrightarrow & x = \frac{34}{9} & & \\ & V = \left\{ \frac{34}{9} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 25x+60 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-25x} & = & \color{red}{25x} +60 \color{blue}{-25x} \color{blue}{-8} \\\Leftrightarrow & -19x& = & 52 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{52}{-19} \\\Leftrightarrow & x = \frac{-52}{19} & & \\ & V = \left\{ \frac{-52}{19} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -126x-90 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+126x} & = & \color{red}{-126x} -90 \color{blue}{+126x} \color{blue}{-20} \\\Leftrightarrow & 141x& = & -110 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-110}{141} \\\Leftrightarrow & x = \frac{-110}{141} & & \\ & V = \left\{ \frac{-110}{141} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-9& = & 21x-126 \\\Leftrightarrow & 14x \color{red}{-9} \color{blue}{+9} \color{blue}{-21x} & = & \color{red}{21x} -126 \color{blue}{-21x} \color{blue}{+9} \\\Leftrightarrow & -7x& = & -117 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-117}{-7} \\\Leftrightarrow & x = \frac{117}{7} & & \\ & V = \left\{ \frac{117}{7} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-182}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & -182x+650 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{+182x} & = & \color{red}{-182x} +650 \color{blue}{+182x} \color{blue}{-20} \\\Leftrightarrow & 247x& = & 630 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{630}{247} \\\Leftrightarrow & x = \frac{630}{247} & & \\ & V = \left\{ \frac{630}{247} \right\} & \\\end{align}\)
10. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{1}{6}x+2 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 84 }{ \color{blue}{462} })& = & (\frac{77}{ \color{blue}{462} }x+\frac{924}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+84& = & 77x+924 \\\Leftrightarrow & 66x \color{red}{+84} \color{blue}{-84} \color{blue}{-77x} & = & \color{red}{77x} +924 \color{blue}{-77x} \color{blue}{-84} \\\Leftrightarrow & -11x& = & 840 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{840}{-11} \\\Leftrightarrow & x = \frac{-840}{11} & & \\ & V = \left\{ \frac{-840}{11} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{5}{4}x+1 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{165}{ \color{blue}{132} }x+\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-60& = & 165x+132 \\\Leftrightarrow & 44x \color{red}{-60} \color{blue}{+60} \color{blue}{-165x} & = & \color{red}{165x} +132 \color{blue}{-165x} \color{blue}{+60} \\\Leftrightarrow & -121x& = & 192 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{192}{-121} \\\Leftrightarrow & x = \frac{-192}{121} & & \\ & V = \left\{ \frac{-192}{121} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 6 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-25& = & 36x+90 \\\Leftrightarrow & 15x \color{red}{-25} \color{blue}{+25} \color{blue}{-36x} & = & \color{red}{36x} +90 \color{blue}{-36x} \color{blue}{+25} \\\Leftrightarrow & -21x& = & 115 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{115}{-21} \\\Leftrightarrow & x = \frac{-115}{21} & & \\ & V = \left\{ \frac{-115}{21} \right\} & \\\end{align}\)