Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }- \frac{ 4 }{ \color{blue}{15} })& = & (\frac{3}{ \color{blue}{15} }x-\frac{15}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x-4& = & 3x-15 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-3x} & = & \color{red}{3x} -15 \color{blue}{-3x} \color{blue}{+4} \\\Leftrightarrow & 2x& = & -11 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{-11}{2} \\\Leftrightarrow & x = \frac{-11}{2} & & \\ & V = \left\{ \frac{-11}{2} \right\} & \\\end{align}\)
2. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{5}{3}x-3 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{325}{ \color{blue}{195} }x-\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+45& = & 325x-585 \\\Leftrightarrow & 39x \color{red}{+45} \color{blue}{-45} \color{blue}{-325x} & = & \color{red}{325x} -585 \color{blue}{-325x} \color{blue}{-45} \\\Leftrightarrow & -286x& = & -630 \\\Leftrightarrow & \frac{-286x}{ \color{red}{-286} }& = & \frac{-630}{-286} \\\Leftrightarrow & x = \frac{315}{143} & & \\ & V = \left\{ \frac{315}{143} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -42x+210 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+42x} & = & \color{red}{-42x} +210 \color{blue}{+42x} \color{blue}{-25} \\\Leftrightarrow & 47x& = & 185 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{185}{47} \\\Leftrightarrow & x = \frac{185}{47} & & \\ & V = \left\{ \frac{185}{47} \right\} & \\\end{align}\)
4. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{5}{6}x+8 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 168 }{ \color{blue}{546} })& = & (\frac{455}{ \color{blue}{546} }x+\frac{4368}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-168& = & 455x+4368 \\\Leftrightarrow & 78x \color{red}{-168} \color{blue}{+168} \color{blue}{-455x} & = & \color{red}{455x} +4368 \color{blue}{-455x} \color{blue}{+168} \\\Leftrightarrow & -377x& = & 4536 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{4536}{-377} \\\Leftrightarrow & x = \frac{-4536}{377} & & \\ & V = \left\{ \frac{-4536}{377} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+30& = & 175x-280 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-175x} & = & \color{red}{175x} -280 \color{blue}{-175x} \color{blue}{-30} \\\Leftrightarrow & -161x& = & -310 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-310}{-161} \\\Leftrightarrow & x = \frac{310}{161} & & \\ & V = \left\{ \frac{310}{161} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x+\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & 48x+1680 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-48x} & = & \color{red}{48x} +1680 \color{blue}{-48x} \color{blue}{+75} \\\Leftrightarrow & -8x& = & 1755 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{1755}{-8} \\\Leftrightarrow & x = \frac{-1755}{8} & & \\ & V = \left\{ \frac{-1755}{8} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & 36x-120 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{-36x} & = & \color{red}{36x} -120 \color{blue}{-36x} \color{blue}{+9} \\\Leftrightarrow & -31x& = & -111 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-111}{-31} \\\Leftrightarrow & x = \frac{111}{31} & & \\ & V = \left\{ \frac{111}{31} \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+6& = & 7x+28 \\\Leftrightarrow & 2x \color{red}{+6} \color{blue}{-6} \color{blue}{-7x} & = & \color{red}{7x} +28 \color{blue}{-7x} \color{blue}{-6} \\\Leftrightarrow & -5x& = & 22 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{22}{-5} \\\Leftrightarrow & x = \frac{-22}{5} & & \\ & V = \left\{ \frac{-22}{5} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 35 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+35& = & 21x-168 \\\Leftrightarrow & 6x \color{red}{+35} \color{blue}{-35} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{-35} \\\Leftrightarrow & -15x& = & -203 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-203}{-15} \\\Leftrightarrow & x = \frac{203}{15} & & \\ & V = \left\{ \frac{203}{15} \right\} & \\\end{align}\)
10. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-252}{ \color{blue}{180} }x-\frac{900}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-40& = & -252x-900 \\\Leftrightarrow & 45x \color{red}{-40} \color{blue}{+40} \color{blue}{+252x} & = & \color{red}{-252x} -900 \color{blue}{+252x} \color{blue}{+40} \\\Leftrightarrow & 297x& = & -860 \\\Leftrightarrow & \frac{297x}{ \color{red}{297} }& = & \frac{-860}{297} \\\Leftrightarrow & x = \frac{-860}{297} & & \\ & V = \left\{ \frac{-860}{297} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x-\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x+45& = & 48x-960 \\\Leftrightarrow & 80x \color{red}{+45} \color{blue}{-45} \color{blue}{-48x} & = & \color{red}{48x} -960 \color{blue}{-48x} \color{blue}{-45} \\\Leftrightarrow & 32x& = & -1005 \\\Leftrightarrow & \frac{32x}{ \color{red}{32} }& = & \frac{-1005}{32} \\\Leftrightarrow & x = \frac{-1005}{32} & & \\ & V = \left\{ \frac{-1005}{32} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{8}{ \color{blue}{24} }x-\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+15& = & 8x-120 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{-8x} & = & \color{red}{8x} -120 \color{blue}{-8x} \color{blue}{-15} \\\Leftrightarrow & -2x& = & -135 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-135}{-2} \\\Leftrightarrow & x = \frac{135}{2} & & \\ & V = \left\{ \frac{135}{2} \right\} & \\\end{align}\)