Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{2}{9}=\frac{-4}{5}x+3\)
2. \(\frac{x}{2}-\frac{4}{7}=\frac{7}{3}x-5\)
3. \(\frac{x}{3}-\frac{4}{15}=\frac{1}{2}x+1\)
4. \(\frac{x}{2}-\frac{3}{13}=\frac{-2}{3}x-6\)
5. \(\frac{x}{3}+\frac{4}{9}=\frac{-7}{5}x-6\)
6. \(\frac{x}{5}+\frac{4}{7}=\frac{1}{6}x-2\)
7. \(\frac{x}{7}+\frac{5}{16}=\frac{7}{4}x+5\)
8. \(\frac{x}{7}+\frac{4}{7}=\frac{7}{5}x+7\)
9. \(\frac{x}{3}-\frac{2}{15}=\frac{3}{2}x-5\)
10. \(\frac{x}{5}-\frac{3}{16}=\frac{1}{6}x+3\)
11. \(\frac{x}{2}-\frac{4}{11}=\frac{4}{3}x-1\)
12. \(\frac{x}{2}+\frac{3}{14}=\frac{-2}{3}x-1\)

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+20& = & -72x+270 \\\Leftrightarrow & 45x \color{red}{+20} \color{blue}{-20} \color{blue}{+72x} & = & \color{red}{-72x} +270 \color{blue}{+72x} \color{blue}{-20} \\\Leftrightarrow & 117x& = & 250 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{250}{117} \\\Leftrightarrow & x = \frac{250}{117} & & \\ & V = \left\{ \frac{250}{117} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 98x-210 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-98x} & = & \color{red}{98x} -210 \color{blue}{-98x} \color{blue}{+24} \\\Leftrightarrow & -77x& = & -186 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-186}{-77} \\\Leftrightarrow & x = \frac{186}{77} & & \\ & V = \left\{ \frac{186}{77} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-8& = & 15x+30 \\\Leftrightarrow & 10x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} +30 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & 38 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{38}{-5} \\\Leftrightarrow & x = \frac{-38}{5} & & \\ & V = \left\{ \frac{-38}{5} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & -52x-468 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{+52x} & = & \color{red}{-52x} -468 \color{blue}{+52x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & -450 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-450}{91} \\\Leftrightarrow & x = \frac{-450}{91} & & \\ & V = \left\{ \frac{-450}{91} \right\} & \\\end{align}\)
5. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }+ \frac{ 20 }{ \color{blue}{45} })& = & (\frac{-63}{ \color{blue}{45} }x-\frac{270}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x+20& = & -63x-270 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+63x} & = & \color{red}{-63x} -270 \color{blue}{+63x} \color{blue}{-20} \\\Leftrightarrow & 78x& = & -290 \\\Leftrightarrow & \frac{78x}{ \color{red}{78} }& = & \frac{-290}{78} \\\Leftrightarrow & x = \frac{-145}{39} & & \\ & V = \left\{ \frac{-145}{39} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{6}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & 35x-420 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{-35x} & = & \color{red}{35x} -420 \color{blue}{-35x} \color{blue}{-120} \\\Leftrightarrow & 7x& = & -540 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-540}{7} \\\Leftrightarrow & x = \frac{-540}{7} & & \\ & V = \left\{ \frac{-540}{7} \right\} & \\\end{align}\)
7. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{7}{4}x+5 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{196}{ \color{blue}{112} }x+\frac{560}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 196x+560 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-196x} & = & \color{red}{196x} +560 \color{blue}{-196x} \color{blue}{-35} \\\Leftrightarrow & -180x& = & 525 \\\Leftrightarrow & \frac{-180x}{ \color{red}{-180} }& = & \frac{525}{-180} \\\Leftrightarrow & x = \frac{-35}{12} & & \\ & V = \left\{ \frac{-35}{12} \right\} & \\\end{align}\)
8. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{7}{5}x+7 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{49}{ \color{blue}{35} }x+\frac{245}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 49x+245 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-49x} & = & \color{red}{49x} +245 \color{blue}{-49x} \color{blue}{-20} \\\Leftrightarrow & -44x& = & 225 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{225}{-44} \\\Leftrightarrow & x = \frac{-225}{44} & & \\ & V = \left\{ \frac{-225}{44} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{3}{2}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{45}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 45x-150 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-45x} & = & \color{red}{45x} -150 \color{blue}{-45x} \color{blue}{+4} \\\Leftrightarrow & -35x& = & -146 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-146}{-35} \\\Leftrightarrow & x = \frac{146}{35} & & \\ & V = \left\{ \frac{146}{35} \right\} & \\\end{align}\)
10. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x+\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & 40x+720 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{-40x} & = & \color{red}{40x} +720 \color{blue}{-40x} \color{blue}{+45} \\\Leftrightarrow & 8x& = & 765 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{765}{8} \\\Leftrightarrow & x = \frac{765}{8} & & \\ & V = \left\{ \frac{765}{8} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{4}{3}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & 88x-66 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{-88x} & = & \color{red}{88x} -66 \color{blue}{-88x} \color{blue}{+24} \\\Leftrightarrow & -55x& = & -42 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-42}{-55} \\\Leftrightarrow & x = \frac{42}{55} & & \\ & V = \left\{ \frac{42}{55} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{14}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+9& = & -28x-42 \\\Leftrightarrow & 21x \color{red}{+9} \color{blue}{-9} \color{blue}{+28x} & = & \color{red}{-28x} -42 \color{blue}{+28x} \color{blue}{-9} \\\Leftrightarrow & 49x& = & -51 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-51}{49} \\\Leftrightarrow & x = \frac{-51}{49} & & \\ & V = \left\{ \frac{-51}{49} \right\} & \\\end{align}\)