Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{7}{6}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{49}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-24& = & 49x+294 \\\Leftrightarrow & 6x \color{red}{-24} \color{blue}{+24} \color{blue}{-49x} & = & \color{red}{49x} +294 \color{blue}{-49x} \color{blue}{+24} \\\Leftrightarrow & -43x& = & 318 \\\Leftrightarrow & \frac{-43x}{ \color{red}{-43} }& = & \frac{318}{-43} \\\Leftrightarrow & x = \frac{-318}{43} & & \\ & V = \left\{ \frac{-318}{43} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{126}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & 126x+540 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{-126x} & = & \color{red}{126x} +540 \color{blue}{-126x} \color{blue}{+20} \\\Leftrightarrow & -81x& = & 560 \\\Leftrightarrow & \frac{-81x}{ \color{red}{-81} }& = & \frac{560}{-81} \\\Leftrightarrow & x = \frac{-560}{81} & & \\ & V = \left\{ \frac{-560}{81} \right\} & \\\end{align}\)
3. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 63 }{ \color{blue}{273} })& = & (\frac{-182}{ \color{blue}{273} }x-\frac{819}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+63& = & -182x-819 \\\Leftrightarrow & 39x \color{red}{+63} \color{blue}{-63} \color{blue}{+182x} & = & \color{red}{-182x} -819 \color{blue}{+182x} \color{blue}{-63} \\\Leftrightarrow & 221x& = & -882 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-882}{221} \\\Leftrightarrow & x = \frac{-882}{221} & & \\ & V = \left\{ \frac{-882}{221} \right\} & \\\end{align}\)
4. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-56& = & 63x+1008 \\\Leftrightarrow & 18x \color{red}{-56} \color{blue}{+56} \color{blue}{-63x} & = & \color{red}{63x} +1008 \color{blue}{-63x} \color{blue}{+56} \\\Leftrightarrow & -45x& = & 1064 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{1064}{-45} \\\Leftrightarrow & x = \frac{-1064}{45} & & \\ & V = \left\{ \frac{-1064}{45} \right\} & \\\end{align}\)
5. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{8}& = & \frac{2}{5}x+3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{48}{ \color{blue}{120} }x+\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+45& = & 48x+360 \\\Leftrightarrow & 20x \color{red}{+45} \color{blue}{-45} \color{blue}{-48x} & = & \color{red}{48x} +360 \color{blue}{-48x} \color{blue}{-45} \\\Leftrightarrow & -28x& = & 315 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{315}{-28} \\\Leftrightarrow & x = \frac{-45}{4} & & \\ & V = \left\{ \frac{-45}{4} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{13}& = & \frac{-2}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-150& = & -156x+2340 \\\Leftrightarrow & 65x \color{red}{-150} \color{blue}{+150} \color{blue}{+156x} & = & \color{red}{-156x} +2340 \color{blue}{+156x} \color{blue}{+150} \\\Leftrightarrow & 221x& = & 2490 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{2490}{221} \\\Leftrightarrow & x = \frac{2490}{221} & & \\ & V = \left\{ \frac{2490}{221} \right\} & \\\end{align}\)
7. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 42 }{ \color{blue}{231} })& = & (\frac{-154}{ \color{blue}{231} }x-\frac{1155}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-42& = & -154x-1155 \\\Leftrightarrow & 33x \color{red}{-42} \color{blue}{+42} \color{blue}{+154x} & = & \color{red}{-154x} -1155 \color{blue}{+154x} \color{blue}{+42} \\\Leftrightarrow & 187x& = & -1113 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-1113}{187} \\\Leftrightarrow & x = \frac{-1113}{187} & & \\ & V = \left\{ \frac{-1113}{187} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & 26x-390 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-26x} & = & \color{red}{26x} -390 \color{blue}{-26x} \color{blue}{+24} \\\Leftrightarrow & 13x& = & -366 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-366}{13} \\\Leftrightarrow & x = \frac{-366}{13} & & \\ & V = \left\{ \frac{-366}{13} \right\} & \\\end{align}\)
9. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 10 }{ \color{blue}{45} })& = & (\frac{105}{ \color{blue}{45} }x-\frac{270}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-10& = & 105x-270 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{-105x} & = & \color{red}{105x} -270 \color{blue}{-105x} \color{blue}{+10} \\\Leftrightarrow & -96x& = & -260 \\\Leftrightarrow & \frac{-96x}{ \color{red}{-96} }& = & \frac{-260}{-96} \\\Leftrightarrow & x = \frac{65}{24} & & \\ & V = \left\{ \frac{65}{24} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & 42x+210 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{-42x} & = & \color{red}{42x} +210 \color{blue}{-42x} \color{blue}{+25} \\\Leftrightarrow & -37x& = & 235 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{235}{-37} \\\Leftrightarrow & x = \frac{-235}{37} & & \\ & V = \left\{ \frac{-235}{37} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 36x+210 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-36x} & = & \color{red}{36x} +210 \color{blue}{-36x} \color{blue}{-4} \\\Leftrightarrow & -31x& = & 206 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{206}{-31} \\\Leftrightarrow & x = \frac{-206}{31} & & \\ & V = \left\{ \frac{-206}{31} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 252x+630 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-252x} & = & \color{red}{252x} +630 \color{blue}{-252x} \color{blue}{+90} \\\Leftrightarrow & -217x& = & 720 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{720}{-217} \\\Leftrightarrow & x = \frac{-720}{217} & & \\ & V = \left\{ \frac{-720}{217} \right\} & \\\end{align}\)