Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 25 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x-\frac{175}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+25& = & -28x-175 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+28x} & = & \color{red}{-28x} -175 \color{blue}{+28x} \color{blue}{-25} \\\Leftrightarrow & 33x& = & -200 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-200}{33} \\\Leftrightarrow & x = \frac{-200}{33} & & \\ & V = \left\{ \frac{-200}{33} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{7}{2}x-8 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x-\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & 245x-560 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{-245x} & = & \color{red}{245x} -560 \color{blue}{-245x} \color{blue}{+21} \\\Leftrightarrow & -235x& = & -539 \\\Leftrightarrow & \frac{-235x}{ \color{red}{-235} }& = & \frac{-539}{-235} \\\Leftrightarrow & x = \frac{539}{235} & & \\ & V = \left\{ \frac{539}{235} \right\} & \\\end{align}\)
3. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 21 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{560}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-21& = & 56x-560 \\\Leftrightarrow & 16x \color{red}{-21} \color{blue}{+21} \color{blue}{-56x} & = & \color{red}{56x} -560 \color{blue}{-56x} \color{blue}{+21} \\\Leftrightarrow & -40x& = & -539 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-539}{-40} \\\Leftrightarrow & x = \frac{539}{40} & & \\ & V = \left\{ \frac{539}{40} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 28x+252 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-28x} & = & \color{red}{28x} +252 \color{blue}{-28x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 228 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{228}{-7} \\\Leftrightarrow & x = \frac{-228}{7} & & \\ & V = \left\{ \frac{-228}{7} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{14}& = & \frac{3}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 30 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-30& = & 63x-504 \\\Leftrightarrow & 28x \color{red}{-30} \color{blue}{+30} \color{blue}{-63x} & = & \color{red}{63x} -504 \color{blue}{-63x} \color{blue}{+30} \\\Leftrightarrow & -35x& = & -474 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-474}{-35} \\\Leftrightarrow & x = \frac{474}{35} & & \\ & V = \left\{ \frac{474}{35} \right\} & \\\end{align}\)
6. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{7}{2}x+6 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{539}{ \color{blue}{154} }x+\frac{924}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 539x+924 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-539x} & = & \color{red}{539x} +924 \color{blue}{-539x} \color{blue}{-28} \\\Leftrightarrow & -517x& = & 896 \\\Leftrightarrow & \frac{-517x}{ \color{red}{-517} }& = & \frac{896}{-517} \\\Leftrightarrow & x = \frac{-896}{517} & & \\ & V = \left\{ \frac{-896}{517} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{14}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 15 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+15& = & 14x-168 \\\Leftrightarrow & 21x \color{red}{+15} \color{blue}{-15} \color{blue}{-14x} & = & \color{red}{14x} -168 \color{blue}{-14x} \color{blue}{-15} \\\Leftrightarrow & 7x& = & -183 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-183}{7} \\\Leftrightarrow & x = \frac{-183}{7} & & \\ & V = \left\{ \frac{-183}{7} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{4}{5}x-5 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{72}{ \color{blue}{90} }x-\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-40& = & 72x-450 \\\Leftrightarrow & 45x \color{red}{-40} \color{blue}{+40} \color{blue}{-72x} & = & \color{red}{72x} -450 \color{blue}{-72x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & -410 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-410}{-27} \\\Leftrightarrow & x = \frac{410}{27} & & \\ & V = \left\{ \frac{410}{27} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{4}{3}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{40}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & 40x-210 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-40x} & = & \color{red}{40x} -210 \color{blue}{-40x} \color{blue}{+9} \\\Leftrightarrow & -34x& = & -201 \\\Leftrightarrow & \frac{-34x}{ \color{red}{-34} }& = & \frac{-201}{-34} \\\Leftrightarrow & x = \frac{201}{34} & & \\ & V = \left\{ \frac{201}{34} \right\} & \\\end{align}\)
10. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{8}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{56}{ \color{blue}{24} }x-\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+9& = & 56x-144 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{-56x} & = & \color{red}{56x} -144 \color{blue}{-56x} \color{blue}{-9} \\\Leftrightarrow & -44x& = & -153 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-153}{-44} \\\Leftrightarrow & x = \frac{153}{44} & & \\ & V = \left\{ \frac{153}{44} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x+\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & -110x+462 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+110x} & = & \color{red}{-110x} +462 \color{blue}{+110x} \color{blue}{-30} \\\Leftrightarrow & 143x& = & 432 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{432}{143} \\\Leftrightarrow & x = \frac{432}{143} & & \\ & V = \left\{ \frac{432}{143} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 40x-400 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-40x} & = & \color{red}{40x} -400 \color{blue}{-40x} \color{blue}{-15} \\\Leftrightarrow & -24x& = & -415 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-415}{-24} \\\Leftrightarrow & x = \frac{415}{24} & & \\ & V = \left\{ \frac{415}{24} \right\} & \\\end{align}\)