Vgln. eerste graad (reeks 4)

Hoofdmenu Eentje per keer  🖨

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

1. \(\frac{x}{7}-\frac{4}{15}=\frac{1}{6}x-3\)
2. \(\frac{x}{6}+\frac{4}{13}=\frac{-4}{5}x-6\)
3. \(\frac{x}{2}+\frac{5}{6}=\frac{-4}{5}x+1\)
4. \(\frac{x}{4}-\frac{2}{13}=\frac{1}{3}x+6\)
5. \(\frac{x}{2}-\frac{5}{8}=\frac{-5}{3}x-3\)
6. \(\frac{x}{2}-\frac{3}{10}=\frac{-5}{3}x+1\)
7. \(\frac{x}{5}+\frac{5}{8}=\frac{-5}{6}x-5\)
8. \(\frac{x}{3}-\frac{4}{11}=\frac{1}{2}x+2\)
9. \(\frac{x}{2}-\frac{4}{9}=\frac{2}{3}x-2\)
10. \(\frac{x}{5}+\frac{5}{16}=\frac{5}{6}x+1\)
11. \(\frac{x}{7}-\frac{2}{7}=\frac{1}{6}x-5\)
12. \(\frac{x}{5}+\frac{4}{9}=\frac{1}{2}x-6\)

---

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 56 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-56& = & 35x-630 \\\Leftrightarrow & 30x \color{red}{-56} \color{blue}{+56} \color{blue}{-35x} & = & \color{red}{35x} -630 \color{blue}{-35x} \color{blue}{+56} \\\Leftrightarrow & -5x& = & -574 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-574}{-5} \\\Leftrightarrow & x = \frac{574}{5} & & \\ & V = \left\{ \frac{574}{5} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -312x-2340 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+312x} & = & \color{red}{-312x} -2340 \color{blue}{+312x} \color{blue}{-120} \\\Leftrightarrow & 377x& = & -2460 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-2460}{377} \\\Leftrightarrow & x = \frac{-2460}{377} & & \\ & V = \left\{ \frac{-2460}{377} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 6 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+25& = & -24x+30 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} +30 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 39x& = & 5 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{5}{39} \\\Leftrightarrow & x = \frac{5}{39} & & \\ & V = \left\{ \frac{5}{39} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x+\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & 52x+936 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-52x} & = & \color{red}{52x} +936 \color{blue}{-52x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & 960 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{960}{-13} \\\Leftrightarrow & x = \frac{-960}{13} & & \\ & V = \left\{ \frac{-960}{13} \right\} & \\\end{align}\)
5. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-40}{ \color{blue}{24} }x-\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & -40x-72 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+40x} & = & \color{red}{-40x} -72 \color{blue}{+40x} \color{blue}{+15} \\\Leftrightarrow & 52x& = & -57 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{-57}{52} \\\Leftrightarrow & x = \frac{-57}{52} & & \\ & V = \left\{ \frac{-57}{52} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & -50x+30 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{+50x} & = & \color{red}{-50x} +30 \color{blue}{+50x} \color{blue}{+9} \\\Leftrightarrow & 65x& = & 39 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{39}{65} \\\Leftrightarrow & x = \frac{3}{5} & & \\ & V = \left\{ \frac{3}{5} \right\} & \\\end{align}\)
7. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-100}{ \color{blue}{120} }x-\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x+75& = & -100x-600 \\\Leftrightarrow & 24x \color{red}{+75} \color{blue}{-75} \color{blue}{+100x} & = & \color{red}{-100x} -600 \color{blue}{+100x} \color{blue}{-75} \\\Leftrightarrow & 124x& = & -675 \\\Leftrightarrow & \frac{124x}{ \color{red}{124} }& = & \frac{-675}{124} \\\Leftrightarrow & x = \frac{-675}{124} & & \\ & V = \left\{ \frac{-675}{124} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-24& = & 33x+132 \\\Leftrightarrow & 22x \color{red}{-24} \color{blue}{+24} \color{blue}{-33x} & = & \color{red}{33x} +132 \color{blue}{-33x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & 156 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{156}{-11} \\\Leftrightarrow & x = \frac{-156}{11} & & \\ & V = \left\{ \frac{-156}{11} \right\} & \\\end{align}\)
9. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{2}{3}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{12}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & 12x-36 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} -36 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & -28 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-28}{-3} \\\Leftrightarrow & x = \frac{28}{3} & & \\ & V = \left\{ \frac{28}{3} \right\} & \\\end{align}\)
10. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{5}{6}x+1 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{200}{ \color{blue}{240} }x+\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & 200x+240 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{-200x} & = & \color{red}{200x} +240 \color{blue}{-200x} \color{blue}{-75} \\\Leftrightarrow & -152x& = & 165 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{165}{-152} \\\Leftrightarrow & x = \frac{-165}{152} & & \\ & V = \left\{ \frac{-165}{152} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{6}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-12& = & 7x-210 \\\Leftrightarrow & 6x \color{red}{-12} \color{blue}{+12} \color{blue}{-7x} & = & \color{red}{7x} -210 \color{blue}{-7x} \color{blue}{+12} \\\Leftrightarrow & -x& = & -198 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-198}{-1} \\\Leftrightarrow & x = 198 & & \\ & V = \left\{ 198 \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 45x-540 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-45x} & = & \color{red}{45x} -540 \color{blue}{-45x} \color{blue}{-40} \\\Leftrightarrow & -27x& = & -580 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-580}{-27} \\\Leftrightarrow & x = \frac{580}{27} & & \\ & V = \left\{ \frac{580}{27} \right\} & \\\end{align}\)