Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }+ \frac{ 35 }{ \color{blue}{63} })& = & (\frac{84}{ \color{blue}{63} }x+\frac{315}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x+35& = & 84x+315 \\\Leftrightarrow & 9x \color{red}{+35} \color{blue}{-35} \color{blue}{-84x} & = & \color{red}{84x} +315 \color{blue}{-84x} \color{blue}{-35} \\\Leftrightarrow & -75x& = & 280 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{280}{-75} \\\Leftrightarrow & x = \frac{-56}{15} & & \\ & V = \left\{ \frac{-56}{15} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & -40x-180 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{+40x} & = & \color{red}{-40x} -180 \color{blue}{+40x} \color{blue}{+16} \\\Leftrightarrow & 55x& = & -164 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{-164}{55} \\\Leftrightarrow & x = \frac{-164}{55} & & \\ & V = \left\{ \frac{-164}{55} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-8}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -112x+336 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+112x} & = & \color{red}{-112x} +336 \color{blue}{+112x} \color{blue}{-12} \\\Leftrightarrow & 133x& = & 324 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{324}{133} \\\Leftrightarrow & x = \frac{324}{133} & & \\ & V = \left\{ \frac{324}{133} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -224x+588 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+224x} & = & \color{red}{-224x} +588 \color{blue}{+224x} \color{blue}{+48} \\\Leftrightarrow & 245x& = & 636 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{636}{245} \\\Leftrightarrow & x = \frac{636}{245} & & \\ & V = \left\{ \frac{636}{245} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+40& = & 35x+70 \\\Leftrightarrow & 14x \color{red}{+40} \color{blue}{-40} \color{blue}{-35x} & = & \color{red}{35x} +70 \color{blue}{-35x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & 30 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{30}{-21} \\\Leftrightarrow & x = \frac{-10}{7} & & \\ & V = \left\{ \frac{-10}{7} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 36x-90 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-36x} & = & \color{red}{36x} -90 \color{blue}{-36x} \color{blue}{+4} \\\Leftrightarrow & -31x& = & -86 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-86}{-31} \\\Leftrightarrow & x = \frac{86}{31} & & \\ & V = \left\{ \frac{86}{31} \right\} & \\\end{align}\)
7. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 66x-330 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-66x} & = & \color{red}{66x} -330 \color{blue}{-66x} \color{blue}{+150} \\\Leftrightarrow & -11x& = & -180 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-180}{-11} \\\Leftrightarrow & x = \frac{180}{11} & & \\ & V = \left\{ \frac{180}{11} \right\} & \\\end{align}\)
8. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-84}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & -84x+336 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{+84x} & = & \color{red}{-84x} +336 \color{blue}{+84x} \color{blue}{+15} \\\Leftrightarrow & 100x& = & 351 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{351}{100} \\\Leftrightarrow & x = \frac{351}{100} & & \\ & V = \left\{ \frac{351}{100} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 252x+210 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-252x} & = & \color{red}{252x} +210 \color{blue}{-252x} \color{blue}{+60} \\\Leftrightarrow & -217x& = & 270 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{270}{-217} \\\Leftrightarrow & x = \frac{-270}{217} & & \\ & V = \left\{ \frac{-270}{217} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & 12x-480 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-12x} & = & \color{red}{12x} -480 \color{blue}{-12x} \color{blue}{-25} \\\Leftrightarrow & -2x& = & -505 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-505}{-2} \\\Leftrightarrow & x = \frac{505}{2} & & \\ & V = \left\{ \frac{505}{2} \right\} & \\\end{align}\)
11. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{5}{3}x+7 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 45 }{ \color{blue}{195} })& = & (\frac{325}{ \color{blue}{195} }x+\frac{1365}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-45& = & 325x+1365 \\\Leftrightarrow & 39x \color{red}{-45} \color{blue}{+45} \color{blue}{-325x} & = & \color{red}{325x} +1365 \color{blue}{-325x} \color{blue}{+45} \\\Leftrightarrow & -286x& = & 1410 \\\Leftrightarrow & \frac{-286x}{ \color{red}{-286} }& = & \frac{1410}{-286} \\\Leftrightarrow & x = \frac{-705}{143} & & \\ & V = \left\{ \frac{-705}{143} \right\} & \\\end{align}\)
12. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x-\frac{180}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-16& = & -27x-180 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{+27x} & = & \color{red}{-27x} -180 \color{blue}{+27x} \color{blue}{+16} \\\Leftrightarrow & 39x& = & -164 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-164}{39} \\\Leftrightarrow & x = \frac{-164}{39} & & \\ & V = \left\{ \frac{-164}{39} \right\} & \\\end{align}\)