Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{13}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 175 }{ \color{blue}{455} })& = & (\frac{-637}{ \color{blue}{455} }x-\frac{1820}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+175& = & -637x-1820 \\\Leftrightarrow & 65x \color{red}{+175} \color{blue}{-175} \color{blue}{+637x} & = & \color{red}{-637x} -1820 \color{blue}{+637x} \color{blue}{-175} \\\Leftrightarrow & 702x& = & -1995 \\\Leftrightarrow & \frac{702x}{ \color{red}{702} }& = & \frac{-1995}{702} \\\Leftrightarrow & x = \frac{-665}{234} & & \\ & V = \left\{ \frac{-665}{234} \right\} & \\\end{align}\)
2. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 168 }{ \color{blue}{462} })& = & (\frac{-385}{ \color{blue}{462} }x-\frac{2310}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-168& = & -385x-2310 \\\Leftrightarrow & 66x \color{red}{-168} \color{blue}{+168} \color{blue}{+385x} & = & \color{red}{-385x} -2310 \color{blue}{+385x} \color{blue}{+168} \\\Leftrightarrow & 451x& = & -2142 \\\Leftrightarrow & \frac{451x}{ \color{red}{451} }& = & \frac{-2142}{451} \\\Leftrightarrow & x = \frac{-2142}{451} & & \\ & V = \left\{ \frac{-2142}{451} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{3}{4}x+8 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{27}{ \color{blue}{36} }x+\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-20& = & 27x+288 \\\Leftrightarrow & 12x \color{red}{-20} \color{blue}{+20} \color{blue}{-27x} & = & \color{red}{27x} +288 \color{blue}{-27x} \color{blue}{+20} \\\Leftrightarrow & -15x& = & 308 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{308}{-15} \\\Leftrightarrow & x = \frac{-308}{15} & & \\ & V = \left\{ \frac{-308}{15} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-12& = & 39x+234 \\\Leftrightarrow & 26x \color{red}{-12} \color{blue}{+12} \color{blue}{-39x} & = & \color{red}{39x} +234 \color{blue}{-39x} \color{blue}{+12} \\\Leftrightarrow & -13x& = & 246 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{246}{-13} \\\Leftrightarrow & x = \frac{-246}{13} & & \\ & V = \left\{ \frac{-246}{13} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-150& = & 65x-2340 \\\Leftrightarrow & 78x \color{red}{-150} \color{blue}{+150} \color{blue}{-65x} & = & \color{red}{65x} -2340 \color{blue}{-65x} \color{blue}{+150} \\\Leftrightarrow & 13x& = & -2190 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-2190}{13} \\\Leftrightarrow & x = \frac{-2190}{13} & & \\ & V = \left\{ \frac{-2190}{13} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 468x-1170 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-468x} & = & \color{red}{468x} -1170 \color{blue}{-468x} \color{blue}{-90} \\\Leftrightarrow & -403x& = & -1260 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-1260}{-403} \\\Leftrightarrow & x = \frac{1260}{403} & & \\ & V = \left\{ \frac{1260}{403} \right\} & \\\end{align}\)
7. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{84}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & 84x+108 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{-84x} & = & \color{red}{84x} +108 \color{blue}{-84x} \color{blue}{+16} \\\Leftrightarrow & -75x& = & 124 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{124}{-75} \\\Leftrightarrow & x = \frac{-124}{75} & & \\ & V = \left\{ \frac{-124}{75} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{-5}{6}x-2 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x-\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & -275x-660 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{+275x} & = & \color{red}{-275x} -660 \color{blue}{+275x} \color{blue}{-120} \\\Leftrightarrow & 341x& = & -780 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{-780}{341} \\\Leftrightarrow & x = \frac{-780}{341} & & \\ & V = \left\{ \frac{-780}{341} \right\} & \\\end{align}\)
9. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{4}{5}x-1 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 80 }{ \color{blue}{220} })& = & (\frac{176}{ \color{blue}{220} }x-\frac{220}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-80& = & 176x-220 \\\Leftrightarrow & 55x \color{red}{-80} \color{blue}{+80} \color{blue}{-176x} & = & \color{red}{176x} -220 \color{blue}{-176x} \color{blue}{+80} \\\Leftrightarrow & -121x& = & -140 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-140}{-121} \\\Leftrightarrow & x = \frac{140}{121} & & \\ & V = \left\{ \frac{140}{121} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{2}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{44}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 44x+66 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-44x} & = & \color{red}{44x} +66 \color{blue}{-44x} \color{blue}{+30} \\\Leftrightarrow & -11x& = & 96 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{96}{-11} \\\Leftrightarrow & x = \frac{-96}{11} & & \\ & V = \left\{ \frac{-96}{11} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & -294x-1680 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{+294x} & = & \color{red}{-294x} -1680 \color{blue}{+294x} \color{blue}{-150} \\\Leftrightarrow & 329x& = & -1830 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-1830}{329} \\\Leftrightarrow & x = \frac{-1830}{329} & & \\ & V = \left\{ \frac{-1830}{329} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{2}{5}x-6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{36}{ \color{blue}{90} }x-\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & 36x-540 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{-36x} & = & \color{red}{36x} -540 \color{blue}{-36x} \color{blue}{+40} \\\Leftrightarrow & -21x& = & -500 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-500}{-21} \\\Leftrightarrow & x = \frac{500}{21} & & \\ & V = \left\{ \frac{500}{21} \right\} & \\\end{align}\)