Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{5}{6}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{325}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+120& = & 325x+2340 \\\Leftrightarrow & 78x \color{red}{+120} \color{blue}{-120} \color{blue}{-325x} & = & \color{red}{325x} +2340 \color{blue}{-325x} \color{blue}{-120} \\\Leftrightarrow & -247x& = & 2220 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{2220}{-247} \\\Leftrightarrow & x = \frac{-2220}{247} & & \\ & V = \left\{ \frac{-2220}{247} \right\} & \\\end{align}\)
2. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 100 }{ \color{blue}{220} })& = & (\frac{-88}{ \color{blue}{220} }x+\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-100& = & -88x+880 \\\Leftrightarrow & 55x \color{red}{-100} \color{blue}{+100} \color{blue}{+88x} & = & \color{red}{-88x} +880 \color{blue}{+88x} \color{blue}{+100} \\\Leftrightarrow & 143x& = & 980 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{980}{143} \\\Leftrightarrow & x = \frac{980}{143} & & \\ & V = \left\{ \frac{980}{143} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{15}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 15x-630 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-15x} & = & \color{red}{15x} -630 \color{blue}{-15x} \color{blue}{+40} \\\Leftrightarrow & 3x& = & -590 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-590}{3} \\\Leftrightarrow & x = \frac{-590}{3} & & \\ & V = \left\{ \frac{-590}{3} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{5}{6}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+18& = & 35x-336 \\\Leftrightarrow & 6x \color{red}{+18} \color{blue}{-18} \color{blue}{-35x} & = & \color{red}{35x} -336 \color{blue}{-35x} \color{blue}{-18} \\\Leftrightarrow & -29x& = & -354 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-354}{-29} \\\Leftrightarrow & x = \frac{354}{29} & & \\ & V = \left\{ \frac{354}{29} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & 16x+240 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} +240 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & 8x& = & 255 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{255}{8} \\\Leftrightarrow & x = \frac{255}{8} & & \\ & V = \left\{ \frac{255}{8} \right\} & \\\end{align}\)
6. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{8}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{72}{ \color{blue}{120} }x+\frac{240}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+45& = & 72x+240 \\\Leftrightarrow & 20x \color{red}{+45} \color{blue}{-45} \color{blue}{-72x} & = & \color{red}{72x} +240 \color{blue}{-72x} \color{blue}{-45} \\\Leftrightarrow & -52x& = & 195 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{195}{-52} \\\Leftrightarrow & x = \frac{-15}{4} & & \\ & V = \left\{ \frac{-15}{4} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -56x+84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+56x} & = & \color{red}{-56x} +84 \color{blue}{+56x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & 108 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{108}{77} \\\Leftrightarrow & x = \frac{108}{77} & & \\ & V = \left\{ \frac{108}{77} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -294x+840 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+294x} & = & \color{red}{-294x} +840 \color{blue}{+294x} \color{blue}{-120} \\\Leftrightarrow & 329x& = & 720 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{720}{329} \\\Leftrightarrow & x = \frac{720}{329} & & \\ & V = \left\{ \frac{720}{329} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 22x+330 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-22x} & = & \color{red}{22x} +330 \color{blue}{-22x} \color{blue}{+12} \\\Leftrightarrow & 11x& = & 342 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{342}{11} \\\Leftrightarrow & x = \frac{342}{11} & & \\ & V = \left\{ \frac{342}{11} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-8}{3}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-208}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -208x-312 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+208x} & = & \color{red}{-208x} -312 \color{blue}{+208x} \color{blue}{-18} \\\Leftrightarrow & 247x& = & -330 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-330}{247} \\\Leftrightarrow & x = \frac{-330}{247} & & \\ & V = \left\{ \frac{-330}{247} \right\} & \\\end{align}\)
11. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{5}{4}x+1 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{45}{ \color{blue}{36} }x+\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+8& = & 45x+36 \\\Leftrightarrow & 12x \color{red}{+8} \color{blue}{-8} \color{blue}{-45x} & = & \color{red}{45x} +36 \color{blue}{-45x} \color{blue}{-8} \\\Leftrightarrow & -33x& = & 28 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{28}{-33} \\\Leftrightarrow & x = \frac{-28}{33} & & \\ & V = \left\{ \frac{-28}{33} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 468x-1950 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-468x} & = & \color{red}{468x} -1950 \color{blue}{-468x} \color{blue}{-90} \\\Leftrightarrow & -403x& = & -2040 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-2040}{-403} \\\Leftrightarrow & x = \frac{2040}{403} & & \\ & V = \left\{ \frac{2040}{403} \right\} & \\\end{align}\)