Vgln. eerste graad (reeks 4)

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

1. \(\frac{x}{5}+\frac{5}{11}=\frac{1}{2}x-8\)
2. \(\frac{x}{3}+\frac{5}{12}=\frac{1}{2}x-4\)
3. \(\frac{x}{6}-\frac{4}{15}=\frac{7}{5}x-2\)
4. \(\frac{x}{7}+\frac{4}{11}=\frac{1}{2}x+2\)
5. \(\frac{x}{6}+\frac{4}{13}=\frac{-4}{5}x-6\)
6. \(\frac{x}{7}+\frac{4}{9}=\frac{1}{2}x+3\)
7. \(\frac{x}{3}-\frac{2}{9}=\frac{1}{2}x+4\)
8. \(\frac{x}{3}+\frac{3}{16}=\frac{7}{2}x-3\)
9. \(\frac{x}{4}+\frac{5}{7}=\frac{7}{3}x+7\)
10. \(\frac{x}{2}-\frac{5}{16}=\frac{1}{3}x+3\)
11. \(\frac{x}{4}-\frac{3}{11}=\frac{-5}{3}x+1\)
12. \(\frac{x}{4}-\frac{5}{6}=\frac{6}{5}x+7\)

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

Verbetersleutel

1. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+50& = & 55x-880 \\\Leftrightarrow & 22x \color{red}{+50} \color{blue}{-50} \color{blue}{-55x} & = & \color{red}{55x} -880 \color{blue}{-55x} \color{blue}{-50} \\\Leftrightarrow & -33x& = & -930 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-930}{-33} \\\Leftrightarrow & x = \frac{310}{11} & & \\ & V = \left\{ \frac{310}{11} \right\} & \\\end{align}\)
2. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{6}{ \color{blue}{12} }x-\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & 6x-48 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{-6x} & = & \color{red}{6x} -48 \color{blue}{-6x} \color{blue}{-5} \\\Leftrightarrow & -2x& = & -53 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-53}{-2} \\\Leftrightarrow & x = \frac{53}{2} & & \\ & V = \left\{ \frac{53}{2} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{7}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 42x-60 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-42x} & = & \color{red}{42x} -60 \color{blue}{-42x} \color{blue}{+8} \\\Leftrightarrow & -37x& = & -52 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{-52}{-37} \\\Leftrightarrow & x = \frac{52}{37} & & \\ & V = \left\{ \frac{52}{37} \right\} & \\\end{align}\)
4. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 56 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{308}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+56& = & 77x+308 \\\Leftrightarrow & 22x \color{red}{+56} \color{blue}{-56} \color{blue}{-77x} & = & \color{red}{77x} +308 \color{blue}{-77x} \color{blue}{-56} \\\Leftrightarrow & -55x& = & 252 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{252}{-55} \\\Leftrightarrow & x = \frac{-252}{55} & & \\ & V = \left\{ \frac{-252}{55} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -312x-2340 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+312x} & = & \color{red}{-312x} -2340 \color{blue}{+312x} \color{blue}{-120} \\\Leftrightarrow & 377x& = & -2460 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-2460}{377} \\\Leftrightarrow & x = \frac{-2460}{377} & & \\ & V = \left\{ \frac{-2460}{377} \right\} & \\\end{align}\)
6. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{378}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 63x+378 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-63x} & = & \color{red}{63x} +378 \color{blue}{-63x} \color{blue}{-56} \\\Leftrightarrow & -45x& = & 322 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{322}{-45} \\\Leftrightarrow & x = \frac{-322}{45} & & \\ & V = \left\{ \frac{-322}{45} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-4& = & 9x+72 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-9x} & = & \color{red}{9x} +72 \color{blue}{-9x} \color{blue}{+4} \\\Leftrightarrow & -3x& = & 76 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{76}{-3} \\\Leftrightarrow & x = \frac{-76}{3} & & \\ & V = \left\{ \frac{-76}{3} \right\} & \\\end{align}\)
8. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{168}{ \color{blue}{48} }x-\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 168x-144 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-168x} & = & \color{red}{168x} -144 \color{blue}{-168x} \color{blue}{-9} \\\Leftrightarrow & -152x& = & -153 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{-153}{-152} \\\Leftrightarrow & x = \frac{153}{152} & & \\ & V = \left\{ \frac{153}{152} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & 196x+588 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{-196x} & = & \color{red}{196x} +588 \color{blue}{-196x} \color{blue}{-60} \\\Leftrightarrow & -175x& = & 528 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{528}{-175} \\\Leftrightarrow & x = \frac{-528}{175} & & \\ & V = \left\{ \frac{-528}{175} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & 16x+144 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} +144 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & 8x& = & 159 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{159}{8} \\\Leftrightarrow & x = \frac{159}{8} & & \\ & V = \left\{ \frac{159}{8} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{11}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 36 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-36& = & -220x+132 \\\Leftrightarrow & 33x \color{red}{-36} \color{blue}{+36} \color{blue}{+220x} & = & \color{red}{-220x} +132 \color{blue}{+220x} \color{blue}{+36} \\\Leftrightarrow & 253x& = & 168 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{168}{253} \\\Leftrightarrow & x = \frac{168}{253} & & \\ & V = \left\{ \frac{168}{253} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{6}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 50 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-50& = & 72x+420 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{-72x} & = & \color{red}{72x} +420 \color{blue}{-72x} \color{blue}{+50} \\\Leftrightarrow & -57x& = & 470 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{470}{-57} \\\Leftrightarrow & x = \frac{-470}{57} & & \\ & V = \left\{ \frac{-470}{57} \right\} & \\\end{align}\)