Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -112x-42 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+112x} & = & \color{red}{-112x} -42 \color{blue}{+112x} \color{blue}{-18} \\\Leftrightarrow & 133x& = & -60 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-60}{133} \\\Leftrightarrow & x = \frac{-60}{133} & & \\ & V = \left\{ \frac{-60}{133} \right\} & \\\end{align}\)
2. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{275}{ \color{blue}{330} }x+\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & 275x+660 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{-275x} & = & \color{red}{275x} +660 \color{blue}{-275x} \color{blue}{-150} \\\Leftrightarrow & -209x& = & 510 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{510}{-209} \\\Leftrightarrow & x = \frac{-510}{209} & & \\ & V = \left\{ \frac{-510}{209} \right\} & \\\end{align}\)
3. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 84 }{ \color{blue}{273} })& = & (\frac{-455}{ \color{blue}{273} }x-\frac{546}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+84& = & -455x-546 \\\Leftrightarrow & 39x \color{red}{+84} \color{blue}{-84} \color{blue}{+455x} & = & \color{red}{-455x} -546 \color{blue}{+455x} \color{blue}{-84} \\\Leftrightarrow & 494x& = & -630 \\\Leftrightarrow & \frac{494x}{ \color{red}{494} }& = & \frac{-630}{494} \\\Leftrightarrow & x = \frac{-315}{247} & & \\ & V = \left\{ \frac{-315}{247} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & -42x-30 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{+42x} & = & \color{red}{-42x} -30 \color{blue}{+42x} \color{blue}{+8} \\\Leftrightarrow & 47x& = & -22 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-22}{47} \\\Leftrightarrow & x = \frac{-22}{47} & & \\ & V = \left\{ \frac{-22}{47} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{3}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 18x+210 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-18x} & = & \color{red}{18x} +210 \color{blue}{-18x} \color{blue}{-4} \\\Leftrightarrow & -13x& = & 206 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{206}{-13} \\\Leftrightarrow & x = \frac{-206}{13} & & \\ & V = \left\{ \frac{-206}{13} \right\} & \\\end{align}\)
6. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{1}{6}x+7 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{21}{ \color{blue}{126} }x+\frac{882}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 21x+882 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-21x} & = & \color{red}{21x} +882 \color{blue}{-21x} \color{blue}{-56} \\\Leftrightarrow & -3x& = & 826 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{826}{-3} \\\Leftrightarrow & x = \frac{-826}{3} & & \\ & V = \left\{ \frac{-826}{3} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 33x-264 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-33x} & = & \color{red}{33x} -264 \color{blue}{-33x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & -288 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-288}{-11} \\\Leftrightarrow & x = \frac{288}{11} & & \\ & V = \left\{ \frac{288}{11} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+24& = & 39x-624 \\\Leftrightarrow & 26x \color{red}{+24} \color{blue}{-24} \color{blue}{-39x} & = & \color{red}{39x} -624 \color{blue}{-39x} \color{blue}{-24} \\\Leftrightarrow & -13x& = & -648 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-648}{-13} \\\Leftrightarrow & x = \frac{648}{13} & & \\ & V = \left\{ \frac{648}{13} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{2}{5}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{24}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 24x+120 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-24x} & = & \color{red}{24x} +120 \color{blue}{-24x} \color{blue}{-8} \\\Leftrightarrow & -9x& = & 112 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{112}{-9} \\\Leftrightarrow & x = \frac{-112}{9} & & \\ & V = \left\{ \frac{-112}{9} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{5}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 105x-294 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-105x} & = & \color{red}{105x} -294 \color{blue}{-105x} \color{blue}{+24} \\\Leftrightarrow & -91x& = & -270 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-270}{-91} \\\Leftrightarrow & x = \frac{270}{91} & & \\ & V = \left\{ \frac{270}{91} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{2}{3}x+3 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{12}{ \color{blue}{18} }x+\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 12x+54 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-12x} & = & \color{red}{12x} +54 \color{blue}{-12x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & 50 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{50}{-3} \\\Leftrightarrow & x = \frac{-50}{3} & & \\ & V = \left\{ \frac{-50}{3} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -294x-1470 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+294x} & = & \color{red}{-294x} -1470 \color{blue}{+294x} \color{blue}{+90} \\\Leftrightarrow & 329x& = & -1380 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-1380}{329} \\\Leftrightarrow & x = \frac{-1380}{329} & & \\ & V = \left\{ \frac{-1380}{329} \right\} & \\\end{align}\)