Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{4}{13}=\frac{-2}{3}x-1\)
2. \(\frac{x}{5}+\frac{5}{14}=\frac{1}{6}x-1\)
3. \(\frac{x}{2}-\frac{3}{14}=\frac{-4}{5}x+3\)
4. \(\frac{x}{3}+\frac{3}{7}=\frac{1}{4}x+2\)
5. \(\frac{x}{6}-\frac{3}{10}=\frac{-4}{5}x-4\)
6. \(\frac{x}{6}+\frac{5}{16}=\frac{6}{5}x+6\)
7. \(\frac{x}{3}-\frac{5}{13}=\frac{1}{2}x-4\)
8. \(\frac{x}{5}+\frac{3}{11}=\frac{-5}{3}x-3\)
9. \(\frac{x}{5}-\frac{5}{11}=\frac{5}{2}x+3\)
10. \(\frac{x}{2}+\frac{2}{11}=\frac{1}{3}x-3\)
11. \(\frac{x}{5}+\frac{3}{13}=\frac{1}{2}x-4\)
12. \(\frac{x}{2}-\frac{5}{9}=\frac{5}{3}x-2\)

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

Verbetersleutel

1. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & -52x-78 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+52x} & = & \color{red}{-52x} -78 \color{blue}{+52x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -102 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-102}{91} \\\Leftrightarrow & x = \frac{-102}{91} & & \\ & V = \left\{ \frac{-102}{91} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+75& = & 35x-210 \\\Leftrightarrow & 42x \color{red}{+75} \color{blue}{-75} \color{blue}{-35x} & = & \color{red}{35x} -210 \color{blue}{-35x} \color{blue}{-75} \\\Leftrightarrow & 7x& = & -285 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-285}{7} \\\Leftrightarrow & x = \frac{-285}{7} & & \\ & V = \left\{ \frac{-285}{7} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & -56x+210 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{+56x} & = & \color{red}{-56x} +210 \color{blue}{+56x} \color{blue}{+15} \\\Leftrightarrow & 91x& = & 225 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{225}{91} \\\Leftrightarrow & x = \frac{225}{91} & & \\ & V = \left\{ \frac{225}{91} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+36& = & 21x+168 \\\Leftrightarrow & 28x \color{red}{+36} \color{blue}{-36} \color{blue}{-21x} & = & \color{red}{21x} +168 \color{blue}{-21x} \color{blue}{-36} \\\Leftrightarrow & 7x& = & 132 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{132}{7} \\\Leftrightarrow & x = \frac{132}{7} & & \\ & V = \left\{ \frac{132}{7} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-4}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -24x-120 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+24x} & = & \color{red}{-24x} -120 \color{blue}{+24x} \color{blue}{+9} \\\Leftrightarrow & 29x& = & -111 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-111}{29} \\\Leftrightarrow & x = \frac{-111}{29} & & \\ & V = \left\{ \frac{-111}{29} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & 288x+1440 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-288x} & = & \color{red}{288x} +1440 \color{blue}{-288x} \color{blue}{-75} \\\Leftrightarrow & -248x& = & 1365 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{1365}{-248} \\\Leftrightarrow & x = \frac{-1365}{248} & & \\ & V = \left\{ \frac{-1365}{248} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-30& = & 39x-312 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-39x} & = & \color{red}{39x} -312 \color{blue}{-39x} \color{blue}{+30} \\\Leftrightarrow & -13x& = & -282 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-282}{-13} \\\Leftrightarrow & x = \frac{282}{13} & & \\ & V = \left\{ \frac{282}{13} \right\} & \\\end{align}\)
8. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 45 }{ \color{blue}{165} })& = & (\frac{-275}{ \color{blue}{165} }x-\frac{495}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+45& = & -275x-495 \\\Leftrightarrow & 33x \color{red}{+45} \color{blue}{-45} \color{blue}{+275x} & = & \color{red}{-275x} -495 \color{blue}{+275x} \color{blue}{-45} \\\Leftrightarrow & 308x& = & -540 \\\Leftrightarrow & \frac{308x}{ \color{red}{308} }& = & \frac{-540}{308} \\\Leftrightarrow & x = \frac{-135}{77} & & \\ & V = \left\{ \frac{-135}{77} \right\} & \\\end{align}\)
9. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{275}{ \color{blue}{110} }x+\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-50& = & 275x+330 \\\Leftrightarrow & 22x \color{red}{-50} \color{blue}{+50} \color{blue}{-275x} & = & \color{red}{275x} +330 \color{blue}{-275x} \color{blue}{+50} \\\Leftrightarrow & -253x& = & 380 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{380}{-253} \\\Leftrightarrow & x = \frac{-380}{253} & & \\ & V = \left\{ \frac{-380}{253} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x-\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & 22x-198 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{-22x} & = & \color{red}{22x} -198 \color{blue}{-22x} \color{blue}{-12} \\\Leftrightarrow & 11x& = & -210 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-210}{11} \\\Leftrightarrow & x = \frac{-210}{11} & & \\ & V = \left\{ \frac{-210}{11} \right\} & \\\end{align}\)
11. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{520}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x-520 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} -520 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & -550 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-550}{-39} \\\Leftrightarrow & x = \frac{550}{39} & & \\ & V = \left\{ \frac{550}{39} \right\} & \\\end{align}\)
12. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{5}{3}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{30}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-10& = & 30x-36 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{-30x} & = & \color{red}{30x} -36 \color{blue}{-30x} \color{blue}{+10} \\\Leftrightarrow & -21x& = & -26 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-26}{-21} \\\Leftrightarrow & x = \frac{26}{21} & & \\ & V = \left\{ \frac{26}{21} \right\} & \\\end{align}\)