Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x-126 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} -126 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & -114 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-114}{-7} \\\Leftrightarrow & x = \frac{114}{7} & & \\ & V = \left\{ \frac{114}{7} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -56x-588 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+56x} & = & \color{red}{-56x} -588 \color{blue}{+56x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & -564 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-564}{77} \\\Leftrightarrow & x = \frac{-564}{77} & & \\ & V = \left\{ \frac{-564}{77} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{245}{ \color{blue}{105} }x-\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+30& = & 245x-735 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-245x} & = & \color{red}{245x} -735 \color{blue}{-245x} \color{blue}{-30} \\\Leftrightarrow & -224x& = & -765 \\\Leftrightarrow & \frac{-224x}{ \color{red}{-224} }& = & \frac{-765}{-224} \\\Leftrightarrow & x = \frac{765}{224} & & \\ & V = \left\{ \frac{765}{224} \right\} & \\\end{align}\)
4. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{21}{ \color{blue}{105} }x-\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+60& = & 21x-420 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-21x} & = & \color{red}{21x} -420 \color{blue}{-21x} \color{blue}{-60} \\\Leftrightarrow & 14x& = & -480 \\\Leftrightarrow & \frac{14x}{ \color{red}{14} }& = & \frac{-480}{14} \\\Leftrightarrow & x = \frac{-240}{7} & & \\ & V = \left\{ \frac{-240}{7} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{-5}{6}x-2 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x-\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & -275x-660 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{+275x} & = & \color{red}{-275x} -660 \color{blue}{+275x} \color{blue}{-120} \\\Leftrightarrow & 341x& = & -780 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{-780}{341} \\\Leftrightarrow & x = \frac{-780}{341} & & \\ & V = \left\{ \frac{-780}{341} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{4}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 21x+504 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-21x} & = & \color{red}{21x} +504 \color{blue}{-21x} \color{blue}{+36} \\\Leftrightarrow & 7x& = & 540 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{540}{7} \\\Leftrightarrow & x = \frac{540}{7} & & \\ & V = \left\{ \frac{540}{7} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & -25x+210 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{+25x} & = & \color{red}{-25x} +210 \color{blue}{+25x} \color{blue}{-25} \\\Leftrightarrow & 31x& = & 185 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{185}{31} \\\Leftrightarrow & x = \frac{185}{31} & & \\ & V = \left\{ \frac{185}{31} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & 20x-60 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{-20x} & = & \color{red}{20x} -60 \color{blue}{-20x} \color{blue}{-16} \\\Leftrightarrow & -5x& = & -76 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-76}{-5} \\\Leftrightarrow & x = \frac{76}{5} & & \\ & V = \left\{ \frac{76}{5} \right\} & \\\end{align}\)
9. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x-\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 48x-720 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-48x} & = & \color{red}{48x} -720 \color{blue}{-48x} \color{blue}{+45} \\\Leftrightarrow & -8x& = & -675 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-675}{-8} \\\Leftrightarrow & x = \frac{675}{8} & & \\ & V = \left\{ \frac{675}{8} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{7}{2}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{735}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & 735x+1680 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{-735x} & = & \color{red}{735x} +1680 \color{blue}{-735x} \color{blue}{-28} \\\Leftrightarrow & -705x& = & 1652 \\\Leftrightarrow & \frac{-705x}{ \color{red}{-705} }& = & \frac{1652}{-705} \\\Leftrightarrow & x = \frac{-1652}{705} & & \\ & V = \left\{ \frac{-1652}{705} \right\} & \\\end{align}\)
11. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{11}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 42 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{616}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+42& = & 77x+616 \\\Leftrightarrow & 22x \color{red}{+42} \color{blue}{-42} \color{blue}{-77x} & = & \color{red}{77x} +616 \color{blue}{-77x} \color{blue}{-42} \\\Leftrightarrow & -55x& = & 574 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{574}{-55} \\\Leftrightarrow & x = \frac{-574}{55} & & \\ & V = \left\{ \frac{-574}{55} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -28x+210 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+28x} & = & \color{red}{-28x} +210 \color{blue}{+28x} \color{blue}{-18} \\\Leftrightarrow & 49x& = & 192 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{192}{49} \\\Leftrightarrow & x = \frac{192}{49} & & \\ & V = \left\{ \frac{192}{49} \right\} & \\\end{align}\)