Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x-\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-15& = & -32x-48 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+32x} & = & \color{red}{-32x} -48 \color{blue}{+32x} \color{blue}{+15} \\\Leftrightarrow & 44x& = & -33 \\\Leftrightarrow & \frac{44x}{ \color{red}{44} }& = & \frac{-33}{44} \\\Leftrightarrow & x = \frac{-3}{4} & & \\ & V = \left\{ \frac{-3}{4} \right\} & \\\end{align}\)
2. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }- \frac{ 6 }{ \color{blue}{20} })& = & (\frac{-16}{ \color{blue}{20} }x+\frac{160}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x-6& = & -16x+160 \\\Leftrightarrow & 5x \color{red}{-6} \color{blue}{+6} \color{blue}{+16x} & = & \color{red}{-16x} +160 \color{blue}{+16x} \color{blue}{+6} \\\Leftrightarrow & 21x& = & 166 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{166}{21} \\\Leftrightarrow & x = \frac{166}{21} & & \\ & V = \left\{ \frac{166}{21} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{7}{2}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{315}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 315x+540 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-315x} & = & \color{red}{315x} +540 \color{blue}{-315x} \color{blue}{-40} \\\Leftrightarrow & -297x& = & 500 \\\Leftrightarrow & \frac{-297x}{ \color{red}{-297} }& = & \frac{500}{-297} \\\Leftrightarrow & x = \frac{-500}{297} & & \\ & V = \left\{ \frac{-500}{297} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -24x-60 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+24x} & = & \color{red}{-24x} -60 \color{blue}{+24x} \color{blue}{+4} \\\Leftrightarrow & 29x& = & -56 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-56}{29} \\\Leftrightarrow & x = \frac{-56}{29} & & \\ & V = \left\{ \frac{-56}{29} \right\} & \\\end{align}\)
5. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }+ \frac{ 56 }{ \color{blue}{308} })& = & (\frac{-231}{ \color{blue}{308} }x+\frac{308}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x+56& = & -231x+308 \\\Leftrightarrow & 44x \color{red}{+56} \color{blue}{-56} \color{blue}{+231x} & = & \color{red}{-231x} +308 \color{blue}{+231x} \color{blue}{-56} \\\Leftrightarrow & 275x& = & 252 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{252}{275} \\\Leftrightarrow & x = \frac{252}{275} & & \\ & V = \left\{ \frac{252}{275} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x+360 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} +360 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & 320 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{320}{-3} \\\Leftrightarrow & x = \frac{-320}{3} & & \\ & V = \left\{ \frac{-320}{3} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & -12x-210 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{+12x} & = & \color{red}{-12x} -210 \color{blue}{+12x} \color{blue}{-8} \\\Leftrightarrow & 17x& = & -218 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-218}{17} \\\Leftrightarrow & x = \frac{-218}{17} & & \\ & V = \left\{ \frac{-218}{17} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{14}& = & \frac{7}{6}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+45& = & 245x-210 \\\Leftrightarrow & 42x \color{red}{+45} \color{blue}{-45} \color{blue}{-245x} & = & \color{red}{245x} -210 \color{blue}{-245x} \color{blue}{-45} \\\Leftrightarrow & -203x& = & -255 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{-255}{-203} \\\Leftrightarrow & x = \frac{255}{203} & & \\ & V = \left\{ \frac{255}{203} \right\} & \\\end{align}\)
9. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 84 }{ \color{blue}{546} })& = & (\frac{91}{ \color{blue}{546} }x-\frac{1638}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-84& = & 91x-1638 \\\Leftrightarrow & 78x \color{red}{-84} \color{blue}{+84} \color{blue}{-91x} & = & \color{red}{91x} -1638 \color{blue}{-91x} \color{blue}{+84} \\\Leftrightarrow & -13x& = & -1554 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-1554}{-13} \\\Leftrightarrow & x = \frac{1554}{13} & & \\ & V = \left\{ \frac{1554}{13} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x+\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-30& = & -52x+312 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{+52x} & = & \color{red}{-52x} +312 \color{blue}{+52x} \color{blue}{+30} \\\Leftrightarrow & 91x& = & 342 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{342}{91} \\\Leftrightarrow & x = \frac{342}{91} & & \\ & V = \left\{ \frac{342}{91} \right\} & \\\end{align}\)
11. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{3}{4}x+6 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{195}{ \color{blue}{260} }x+\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-60& = & 195x+1560 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{-195x} & = & \color{red}{195x} +1560 \color{blue}{-195x} \color{blue}{+60} \\\Leftrightarrow & -143x& = & 1620 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{1620}{-143} \\\Leftrightarrow & x = \frac{-1620}{143} & & \\ & V = \left\{ \frac{-1620}{143} \right\} & \\\end{align}\)
12. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{3}{2}x+5 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{15}{ \color{blue}{10} }x+\frac{50}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x+3& = & 15x+50 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-15x} & = & \color{red}{15x} +50 \color{blue}{-15x} \color{blue}{-3} \\\Leftrightarrow & -13x& = & 47 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{47}{-13} \\\Leftrightarrow & x = \frac{-47}{13} & & \\ & V = \left\{ \frac{-47}{13} \right\} & \\\end{align}\)