Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & -70x+210 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{+70x} & = & \color{red}{-70x} +210 \color{blue}{+70x} \color{blue}{+9} \\\Leftrightarrow & 91x& = & 219 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{219}{91} \\\Leftrightarrow & x = \frac{219}{91} & & \\ & V = \left\{ \frac{219}{91} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{195}{ \color{blue}{78} }x-\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-18& = & 195x-624 \\\Leftrightarrow & 26x \color{red}{-18} \color{blue}{+18} \color{blue}{-195x} & = & \color{red}{195x} -624 \color{blue}{-195x} \color{blue}{+18} \\\Leftrightarrow & -169x& = & -606 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-606}{-169} \\\Leftrightarrow & x = \frac{606}{169} & & \\ & V = \left\{ \frac{606}{169} \right\} & \\\end{align}\)
3. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{9}{ \color{blue}{45} }x+\frac{360}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x-20& = & 9x+360 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-9x} & = & \color{red}{9x} +360 \color{blue}{-9x} \color{blue}{+20} \\\Leftrightarrow & 6x& = & 380 \\\Leftrightarrow & \frac{6x}{ \color{red}{6} }& = & \frac{380}{6} \\\Leftrightarrow & x = \frac{190}{3} & & \\ & V = \left\{ \frac{190}{3} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{275}{ \color{blue}{110} }x+\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+20& = & 275x+330 \\\Leftrightarrow & 22x \color{red}{+20} \color{blue}{-20} \color{blue}{-275x} & = & \color{red}{275x} +330 \color{blue}{-275x} \color{blue}{-20} \\\Leftrightarrow & -253x& = & 310 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{310}{-253} \\\Leftrightarrow & x = \frac{-310}{253} & & \\ & V = \left\{ \frac{-310}{253} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{9}& = & \frac{3}{5}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{54}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-20& = & 54x+270 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-54x} & = & \color{red}{54x} +270 \color{blue}{-54x} \color{blue}{+20} \\\Leftrightarrow & -39x& = & 290 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{290}{-39} \\\Leftrightarrow & x = \frac{-290}{39} & & \\ & V = \left\{ \frac{-290}{39} \right\} & \\\end{align}\)
6. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{8}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x+\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-45& = & -168x+480 \\\Leftrightarrow & 20x \color{red}{-45} \color{blue}{+45} \color{blue}{+168x} & = & \color{red}{-168x} +480 \color{blue}{+168x} \color{blue}{+45} \\\Leftrightarrow & 188x& = & 525 \\\Leftrightarrow & \frac{188x}{ \color{red}{188} }& = & \frac{525}{188} \\\Leftrightarrow & x = \frac{525}{188} & & \\ & V = \left\{ \frac{525}{188} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{4}{3}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{88}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & 88x-264 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{-88x} & = & \color{red}{88x} -264 \color{blue}{-88x} \color{blue}{-30} \\\Leftrightarrow & -55x& = & -294 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-294}{-55} \\\Leftrightarrow & x = \frac{294}{55} & & \\ & V = \left\{ \frac{294}{55} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{2}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & 84x-840 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{-84x} & = & \color{red}{84x} -840 \color{blue}{-84x} \color{blue}{+150} \\\Leftrightarrow & -49x& = & -690 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-690}{-49} \\\Leftrightarrow & x = \frac{690}{49} & & \\ & V = \left\{ \frac{690}{49} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{231}{ \color{blue}{66} }x+\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 231x+462 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-231x} & = & \color{red}{231x} +462 \color{blue}{-231x} \color{blue}{-30} \\\Leftrightarrow & -209x& = & 432 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{432}{-209} \\\Leftrightarrow & x = \frac{-432}{209} & & \\ & V = \left\{ \frac{-432}{209} \right\} & \\\end{align}\)
10. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 30 }{ \color{blue}{110} })& = & (\frac{275}{ \color{blue}{110} }x-\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+30& = & 275x-660 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-275x} & = & \color{red}{275x} -660 \color{blue}{-275x} \color{blue}{-30} \\\Leftrightarrow & -253x& = & -690 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{-690}{-253} \\\Leftrightarrow & x = \frac{30}{11} & & \\ & V = \left\{ \frac{30}{11} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -312x+2340 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+312x} & = & \color{red}{-312x} +2340 \color{blue}{+312x} \color{blue}{-90} \\\Leftrightarrow & 377x& = & 2250 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{2250}{377} \\\Leftrightarrow & x = \frac{2250}{377} & & \\ & V = \left\{ \frac{2250}{377} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-24& = & 39x+312 \\\Leftrightarrow & 26x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} +312 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & 336 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{336}{-13} \\\Leftrightarrow & x = \frac{-336}{13} & & \\ & V = \left\{ \frac{-336}{13} \right\} & \\\end{align}\)