Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 28 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{728}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+28& = & 91x+728 \\\Leftrightarrow & 26x \color{red}{+28} \color{blue}{-28} \color{blue}{-91x} & = & \color{red}{91x} +728 \color{blue}{-91x} \color{blue}{-28} \\\Leftrightarrow & -65x& = & 700 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{700}{-65} \\\Leftrightarrow & x = \frac{-140}{13} & & \\ & V = \left\{ \frac{-140}{13} \right\} & \\\end{align}\)
2. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{-231}{ \color{blue}{165} }x-\frac{825}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-75& = & -231x-825 \\\Leftrightarrow & 55x \color{red}{-75} \color{blue}{+75} \color{blue}{+231x} & = & \color{red}{-231x} -825 \color{blue}{+231x} \color{blue}{+75} \\\Leftrightarrow & 286x& = & -750 \\\Leftrightarrow & \frac{286x}{ \color{red}{286} }& = & \frac{-750}{286} \\\Leftrightarrow & x = \frac{-375}{143} & & \\ & V = \left\{ \frac{-375}{143} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & -42x+120 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{+42x} & = & \color{red}{-42x} +120 \color{blue}{+42x} \color{blue}{+4} \\\Leftrightarrow & 57x& = & 124 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{124}{57} \\\Leftrightarrow & x = \frac{124}{57} & & \\ & V = \left\{ \frac{124}{57} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & 42x+630 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{-42x} & = & \color{red}{42x} +630 \color{blue}{-42x} \color{blue}{-45} \\\Leftrightarrow & -7x& = & 585 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{585}{-7} \\\Leftrightarrow & x = \frac{-585}{7} & & \\ & V = \left\{ \frac{-585}{7} \right\} & \\\end{align}\)
5. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{8}& = & \frac{5}{2}x-5 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }- \frac{ 35 }{ \color{blue}{56} })& = & (\frac{140}{ \color{blue}{56} }x-\frac{280}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x-35& = & 140x-280 \\\Leftrightarrow & 8x \color{red}{-35} \color{blue}{+35} \color{blue}{-140x} & = & \color{red}{140x} -280 \color{blue}{-140x} \color{blue}{+35} \\\Leftrightarrow & -132x& = & -245 \\\Leftrightarrow & \frac{-132x}{ \color{red}{-132} }& = & \frac{-245}{-132} \\\Leftrightarrow & x = \frac{245}{132} & & \\ & V = \left\{ \frac{245}{132} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{7}{4}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{105}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+18& = & 105x-480 \\\Leftrightarrow & 20x \color{red}{+18} \color{blue}{-18} \color{blue}{-105x} & = & \color{red}{105x} -480 \color{blue}{-105x} \color{blue}{-18} \\\Leftrightarrow & -85x& = & -498 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{-498}{-85} \\\Leftrightarrow & x = \frac{498}{85} & & \\ & V = \left\{ \frac{498}{85} \right\} & \\\end{align}\)
7. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{26}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+40& = & 26x-130 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{-26x} & = & \color{red}{26x} -130 \color{blue}{-26x} \color{blue}{-40} \\\Leftrightarrow & 39x& = & -170 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-170}{39} \\\Leftrightarrow & x = \frac{-170}{39} & & \\ & V = \left\{ \frac{-170}{39} \right\} & \\\end{align}\)
8. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 80 }{ \color{blue}{180} })& = & (\frac{-252}{ \color{blue}{180} }x-\frac{900}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-80& = & -252x-900 \\\Leftrightarrow & 45x \color{red}{-80} \color{blue}{+80} \color{blue}{+252x} & = & \color{red}{-252x} -900 \color{blue}{+252x} \color{blue}{+80} \\\Leftrightarrow & 297x& = & -820 \\\Leftrightarrow & \frac{297x}{ \color{red}{297} }& = & \frac{-820}{297} \\\Leftrightarrow & x = \frac{-820}{297} & & \\ & V = \left\{ \frac{-820}{297} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 30 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+30& = & 7x-168 \\\Leftrightarrow & 6x \color{red}{+30} \color{blue}{-30} \color{blue}{-7x} & = & \color{red}{7x} -168 \color{blue}{-7x} \color{blue}{-30} \\\Leftrightarrow & -x& = & -198 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-198}{-1} \\\Leftrightarrow & x = 198 & & \\ & V = \left\{ 198 \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 22x+132 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-22x} & = & \color{red}{22x} +132 \color{blue}{-22x} \color{blue}{+30} \\\Leftrightarrow & 11x& = & 162 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{162}{11} \\\Leftrightarrow & x = \frac{162}{11} & & \\ & V = \left\{ \frac{162}{11} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 78x-1950 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-78x} & = & \color{red}{78x} -1950 \color{blue}{-78x} \color{blue}{+120} \\\Leftrightarrow & -13x& = & -1830 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-1830}{-13} \\\Leftrightarrow & x = \frac{1830}{13} & & \\ & V = \left\{ \frac{1830}{13} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 63 }{ \color{blue}{210} })& = & (\frac{70}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-63& = & 70x+1470 \\\Leftrightarrow & 30x \color{red}{-63} \color{blue}{+63} \color{blue}{-70x} & = & \color{red}{70x} +1470 \color{blue}{-70x} \color{blue}{+63} \\\Leftrightarrow & -40x& = & 1533 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{1533}{-40} \\\Leftrightarrow & x = \frac{-1533}{40} & & \\ & V = \left\{ \frac{-1533}{40} \right\} & \\\end{align}\)