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

---

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 10x+210 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-10x} & = & \color{red}{10x} +210 \color{blue}{-10x} \color{blue}{-8} \\\Leftrightarrow & 5x& = & 202 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{202}{5} \\\Leftrightarrow & x = \frac{202}{5} & & \\ & V = \left\{ \frac{202}{5} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 42x+1680 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-42x} & = & \color{red}{42x} +1680 \color{blue}{-42x} \color{blue}{-90} \\\Leftrightarrow & -7x& = & 1590 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{1590}{-7} \\\Leftrightarrow & x = \frac{-1590}{7} & & \\ & V = \left\{ \frac{-1590}{7} \right\} & \\\end{align}\)
3. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & 52x+624 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{-52x} & = & \color{red}{52x} +624 \color{blue}{-52x} \color{blue}{+36} \\\Leftrightarrow & -13x& = & 660 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{660}{-13} \\\Leftrightarrow & x = \frac{-660}{13} & & \\ & V = \left\{ \frac{-660}{13} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & -224x-672 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{+224x} & = & \color{red}{-224x} -672 \color{blue}{+224x} \color{blue}{+36} \\\Leftrightarrow & 245x& = & -636 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-636}{245} \\\Leftrightarrow & x = \frac{-636}{245} & & \\ & V = \left\{ \frac{-636}{245} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{7}{5}x+3 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{147}{ \color{blue}{105} }x+\frac{315}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+30& = & 147x+315 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-147x} & = & \color{red}{147x} +315 \color{blue}{-147x} \color{blue}{-30} \\\Leftrightarrow & -112x& = & 285 \\\Leftrightarrow & \frac{-112x}{ \color{red}{-112} }& = & \frac{285}{-112} \\\Leftrightarrow & x = \frac{-285}{112} & & \\ & V = \left\{ \frac{-285}{112} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & -84x-480 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{+84x} & = & \color{red}{-84x} -480 \color{blue}{+84x} \color{blue}{-25} \\\Leftrightarrow & 94x& = & -505 \\\Leftrightarrow & \frac{94x}{ \color{red}{94} }& = & \frac{-505}{94} \\\Leftrightarrow & x = \frac{-505}{94} & & \\ & V = \left\{ \frac{-505}{94} \right\} & \\\end{align}\)
7. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 20 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-20& = & 55x-440 \\\Leftrightarrow & 22x \color{red}{-20} \color{blue}{+20} \color{blue}{-55x} & = & \color{red}{55x} -440 \color{blue}{-55x} \color{blue}{+20} \\\Leftrightarrow & -33x& = & -420 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-420}{-33} \\\Leftrightarrow & x = \frac{140}{11} & & \\ & V = \left\{ \frac{140}{11} \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{3}{2}x+7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 3 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x+\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+3& = & 21x+98 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-21x} & = & \color{red}{21x} +98 \color{blue}{-21x} \color{blue}{-3} \\\Leftrightarrow & -19x& = & 95 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{95}{-19} \\\Leftrightarrow & x = -5 & & \\ & V = \left\{ -5 \right\} & \\\end{align}\)
9. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 20 }{ \color{blue}{110} })& = & (\frac{132}{ \color{blue}{110} }x+\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+20& = & 132x+110 \\\Leftrightarrow & 55x \color{red}{+20} \color{blue}{-20} \color{blue}{-132x} & = & \color{red}{132x} +110 \color{blue}{-132x} \color{blue}{-20} \\\Leftrightarrow & -77x& = & 90 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{90}{-77} \\\Leftrightarrow & x = \frac{-90}{77} & & \\ & V = \left\{ \frac{-90}{77} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & 468x+1950 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{-468x} & = & \color{red}{468x} +1950 \color{blue}{-468x} \color{blue}{-120} \\\Leftrightarrow & -403x& = & 1830 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{1830}{-403} \\\Leftrightarrow & x = \frac{-1830}{403} & & \\ & V = \left\{ \frac{-1830}{403} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-4& = & 9x-18 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-9x} & = & \color{red}{9x} -18 \color{blue}{-9x} \color{blue}{+4} \\\Leftrightarrow & -3x& = & -14 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-14}{-3} \\\Leftrightarrow & x = \frac{14}{3} & & \\ & V = \left\{ \frac{14}{3} \right\} & \\\end{align}\)
12. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-195}{ \color{blue}{260} }x+\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-80& = & -195x+780 \\\Leftrightarrow & 52x \color{red}{-80} \color{blue}{+80} \color{blue}{+195x} & = & \color{red}{-195x} +780 \color{blue}{+195x} \color{blue}{+80} \\\Leftrightarrow & 247x& = & 860 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{860}{247} \\\Leftrightarrow & x = \frac{860}{247} & & \\ & V = \left\{ \frac{860}{247} \right\} & \\\end{align}\)