Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & -260x-780 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+260x} & = & \color{red}{-260x} -780 \color{blue}{+260x} \color{blue}{+24} \\\Leftrightarrow & 299x& = & -756 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{-756}{299} \\\Leftrightarrow & x = \frac{-756}{299} & & \\ & V = \left\{ \frac{-756}{299} \right\} & \\\end{align}\)
2. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{14}{ \color{blue}{6} }x-\frac{6}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 14x-6 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-14x} & = & \color{red}{14x} -6 \color{blue}{-14x} \color{blue}{-5} \\\Leftrightarrow & -11x& = & -11 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-11}{-11} \\\Leftrightarrow & x = 1 & & \\ & V = \left\{ 1 \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{5}{4}x-8 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{195}{ \color{blue}{156} }x-\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-24& = & 195x-1248 \\\Leftrightarrow & 52x \color{red}{-24} \color{blue}{+24} \color{blue}{-195x} & = & \color{red}{195x} -1248 \color{blue}{-195x} \color{blue}{+24} \\\Leftrightarrow & -143x& = & -1224 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-1224}{-143} \\\Leftrightarrow & x = \frac{1224}{143} & & \\ & V = \left\{ \frac{1224}{143} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+30& = & 14x-280 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-14x} & = & \color{red}{14x} -280 \color{blue}{-14x} \color{blue}{-30} \\\Leftrightarrow & 21x& = & -310 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{-310}{21} \\\Leftrightarrow & x = \frac{-310}{21} & & \\ & V = \left\{ \frac{-310}{21} \right\} & \\\end{align}\)
5. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{5}{2}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{325}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+20& = & 325x+650 \\\Leftrightarrow & 26x \color{red}{+20} \color{blue}{-20} \color{blue}{-325x} & = & \color{red}{325x} +650 \color{blue}{-325x} \color{blue}{-20} \\\Leftrightarrow & -299x& = & 630 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{630}{-299} \\\Leftrightarrow & x = \frac{-630}{299} & & \\ & V = \left\{ \frac{-630}{299} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & 36x-120 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{-36x} & = & \color{red}{36x} -120 \color{blue}{-36x} \color{blue}{+9} \\\Leftrightarrow & -31x& = & -111 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-111}{-31} \\\Leftrightarrow & x = \frac{111}{31} & & \\ & V = \left\{ \frac{111}{31} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x+\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & 26x+546 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{-26x} & = & \color{red}{26x} +546 \color{blue}{-26x} \color{blue}{-12} \\\Leftrightarrow & 13x& = & 534 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{534}{13} \\\Leftrightarrow & x = \frac{534}{13} & & \\ & V = \left\{ \frac{534}{13} \right\} & \\\end{align}\)
8. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{132}{ \color{blue}{110} }x+\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+50& = & 132x+880 \\\Leftrightarrow & 55x \color{red}{+50} \color{blue}{-50} \color{blue}{-132x} & = & \color{red}{132x} +880 \color{blue}{-132x} \color{blue}{-50} \\\Leftrightarrow & -77x& = & 830 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{830}{-77} \\\Leftrightarrow & x = \frac{-830}{77} & & \\ & V = \left\{ \frac{-830}{77} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 20x+180 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-20x} & = & \color{red}{20x} +180 \color{blue}{-20x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & 172 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{172}{-5} \\\Leftrightarrow & x = \frac{-172}{5} & & \\ & V = \left\{ \frac{-172}{5} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-336 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -336 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -312 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-312}{-7} \\\Leftrightarrow & x = \frac{312}{7} & & \\ & V = \left\{ \frac{312}{7} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 20x+120 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-20x} & = & \color{red}{20x} +120 \color{blue}{-20x} \color{blue}{+16} \\\Leftrightarrow & -5x& = & 136 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{136}{-5} \\\Leftrightarrow & x = \frac{-136}{5} & & \\ & V = \left\{ \frac{-136}{5} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{156}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 156x+3120 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-156x} & = & \color{red}{156x} +3120 \color{blue}{-156x} \color{blue}{+120} \\\Leftrightarrow & -91x& = & 3240 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{3240}{-91} \\\Leftrightarrow & x = \frac{-3240}{91} & & \\ & V = \left\{ \frac{-3240}{91} \right\} & \\\end{align}\)