Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x+280 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} +280 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & 260 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{260}{-21} \\\Leftrightarrow & x = \frac{-260}{21} & & \\ & V = \left\{ \frac{-260}{21} \right\} & \\\end{align}\)
2. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }+ \frac{ 28 }{ \color{blue}{63} })& = & (\frac{-42}{ \color{blue}{63} }x-\frac{189}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x+28& = & -42x-189 \\\Leftrightarrow & 9x \color{red}{+28} \color{blue}{-28} \color{blue}{+42x} & = & \color{red}{-42x} -189 \color{blue}{+42x} \color{blue}{-28} \\\Leftrightarrow & 51x& = & -217 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-217}{51} \\\Leftrightarrow & x = \frac{-217}{51} & & \\ & V = \left\{ \frac{-217}{51} \right\} & \\\end{align}\)
3. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{-8}{3}x-5 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 30 }{ \color{blue}{195} })& = & (\frac{-520}{ \color{blue}{195} }x-\frac{975}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-30& = & -520x-975 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{+520x} & = & \color{red}{-520x} -975 \color{blue}{+520x} \color{blue}{+30} \\\Leftrightarrow & 559x& = & -945 \\\Leftrightarrow & \frac{559x}{ \color{red}{559} }& = & \frac{-945}{559} \\\Leftrightarrow & x = \frac{-945}{559} & & \\ & V = \left\{ \frac{-945}{559} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -140x-336 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+140x} & = & \color{red}{-140x} -336 \color{blue}{+140x} \color{blue}{+48} \\\Leftrightarrow & 161x& = & -288 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-288}{161} \\\Leftrightarrow & x = \frac{-288}{161} & & \\ & V = \left\{ \frac{-288}{161} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & -220x+660 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{+220x} & = & \color{red}{-220x} +660 \color{blue}{+220x} \color{blue}{-60} \\\Leftrightarrow & 253x& = & 600 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{600}{253} \\\Leftrightarrow & x = \frac{600}{253} & & \\ & V = \left\{ \frac{600}{253} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x-84 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} -84 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & -96 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-96}{-7} \\\Leftrightarrow & x = \frac{96}{7} & & \\ & V = \left\{ \frac{96}{7} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{165}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 165x-264 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-165x} & = & \color{red}{165x} -264 \color{blue}{-165x} \color{blue}{-24} \\\Leftrightarrow & -143x& = & -288 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-288}{-143} \\\Leftrightarrow & x = \frac{288}{143} & & \\ & V = \left\{ \frac{288}{143} \right\} & \\\end{align}\)
8. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }- \frac{ 10 }{ \color{blue}{45} })& = & (\frac{-18}{ \color{blue}{45} }x-\frac{270}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x-10& = & -18x-270 \\\Leftrightarrow & 15x \color{red}{-10} \color{blue}{+10} \color{blue}{+18x} & = & \color{red}{-18x} -270 \color{blue}{+18x} \color{blue}{+10} \\\Leftrightarrow & 33x& = & -260 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-260}{33} \\\Leftrightarrow & x = \frac{-260}{33} & & \\ & V = \left\{ \frac{-260}{33} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-9& = & -32x+288 \\\Leftrightarrow & 12x \color{red}{-9} \color{blue}{+9} \color{blue}{+32x} & = & \color{red}{-32x} +288 \color{blue}{+32x} \color{blue}{+9} \\\Leftrightarrow & 44x& = & 297 \\\Leftrightarrow & \frac{44x}{ \color{red}{44} }& = & \frac{297}{44} \\\Leftrightarrow & x = \frac{27}{4} & & \\ & V = \left\{ \frac{27}{4} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+12& = & 33x-198 \\\Leftrightarrow & 22x \color{red}{+12} \color{blue}{-12} \color{blue}{-33x} & = & \color{red}{33x} -198 \color{blue}{-33x} \color{blue}{-12} \\\Leftrightarrow & -11x& = & -210 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-210}{-11} \\\Leftrightarrow & x = \frac{210}{11} & & \\ & V = \left\{ \frac{210}{11} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{2}{5}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{24}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & 24x-240 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-24x} & = & \color{red}{24x} -240 \color{blue}{-24x} \color{blue}{-25} \\\Leftrightarrow & -14x& = & -265 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{-265}{-14} \\\Leftrightarrow & x = \frac{265}{14} & & \\ & V = \left\{ \frac{265}{14} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & -35x+252 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{+35x} & = & \color{red}{-35x} +252 \color{blue}{+35x} \color{blue}{+35} \\\Leftrightarrow & 41x& = & 287 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{287}{41} \\\Leftrightarrow & x = 7 & & \\ & V = \left\{ 7 \right\} & \\\end{align}\)