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

---

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{5}{3}x-5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{110}{ \color{blue}{66} }x-\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 110x-330 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-110x} & = & \color{red}{110x} -330 \color{blue}{-110x} \color{blue}{+12} \\\Leftrightarrow & -77x& = & -318 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-318}{-77} \\\Leftrightarrow & x = \frac{318}{77} & & \\ & V = \left\{ \frac{318}{77} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{13}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 30 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+30& = & -130x-312 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{+130x} & = & \color{red}{-130x} -312 \color{blue}{+130x} \color{blue}{-30} \\\Leftrightarrow & 169x& = & -342 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-342}{169} \\\Leftrightarrow & x = \frac{-342}{169} & & \\ & V = \left\{ \frac{-342}{169} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 22x+66 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-22x} & = & \color{red}{22x} +66 \color{blue}{-22x} \color{blue}{+30} \\\Leftrightarrow & 11x& = & 96 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{96}{11} \\\Leftrightarrow & x = \frac{96}{11} & & \\ & V = \left\{ \frac{96}{11} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 21x-504 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-21x} & = & \color{red}{21x} -504 \color{blue}{-21x} \color{blue}{+36} \\\Leftrightarrow & 7x& = & -468 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-468}{7} \\\Leftrightarrow & x = \frac{-468}{7} & & \\ & V = \left\{ \frac{-468}{7} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x+\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & -192x+480 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+192x} & = & \color{red}{-192x} +480 \color{blue}{+192x} \color{blue}{-45} \\\Leftrightarrow & 232x& = & 435 \\\Leftrightarrow & \frac{232x}{ \color{red}{232} }& = & \frac{435}{232} \\\Leftrightarrow & x = \frac{15}{8} & & \\ & V = \left\{ \frac{15}{8} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -168x+1680 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+168x} & = & \color{red}{-168x} +1680 \color{blue}{+168x} \color{blue}{-120} \\\Leftrightarrow & 203x& = & 1560 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{1560}{203} \\\Leftrightarrow & x = \frac{1560}{203} & & \\ & V = \left\{ \frac{1560}{203} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+20& = & -72x-720 \\\Leftrightarrow & 45x \color{red}{+20} \color{blue}{-20} \color{blue}{+72x} & = & \color{red}{-72x} -720 \color{blue}{+72x} \color{blue}{-20} \\\Leftrightarrow & 117x& = & -740 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-740}{117} \\\Leftrightarrow & x = \frac{-740}{117} & & \\ & V = \left\{ \frac{-740}{117} \right\} & \\\end{align}\)
8. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 100 }{ \color{blue}{220} })& = & (\frac{-176}{ \color{blue}{220} }x+\frac{1540}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+100& = & -176x+1540 \\\Leftrightarrow & 55x \color{red}{+100} \color{blue}{-100} \color{blue}{+176x} & = & \color{red}{-176x} +1540 \color{blue}{+176x} \color{blue}{-100} \\\Leftrightarrow & 231x& = & 1440 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{1440}{231} \\\Leftrightarrow & x = \frac{480}{77} & & \\ & V = \left\{ \frac{480}{77} \right\} & \\\end{align}\)
9. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{2}{3}x-7 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }- \frac{ 2 }{ \color{blue}{15} })& = & (\frac{10}{ \color{blue}{15} }x-\frac{105}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x-2& = & 10x-105 \\\Leftrightarrow & 3x \color{red}{-2} \color{blue}{+2} \color{blue}{-10x} & = & \color{red}{10x} -105 \color{blue}{-10x} \color{blue}{+2} \\\Leftrightarrow & -7x& = & -103 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-103}{-7} \\\Leftrightarrow & x = \frac{103}{7} & & \\ & V = \left\{ \frac{103}{7} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+20& = & 45x-450 \\\Leftrightarrow & 18x \color{red}{+20} \color{blue}{-20} \color{blue}{-45x} & = & \color{red}{45x} -450 \color{blue}{-45x} \color{blue}{-20} \\\Leftrightarrow & -27x& = & -470 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-470}{-27} \\\Leftrightarrow & x = \frac{470}{27} & & \\ & V = \left\{ \frac{470}{27} \right\} & \\\end{align}\)
11. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{11}& = & \frac{7}{2}x-4 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 42 }{ \color{blue}{154} })& = & (\frac{539}{ \color{blue}{154} }x-\frac{616}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+42& = & 539x-616 \\\Leftrightarrow & 22x \color{red}{+42} \color{blue}{-42} \color{blue}{-539x} & = & \color{red}{539x} -616 \color{blue}{-539x} \color{blue}{-42} \\\Leftrightarrow & -517x& = & -658 \\\Leftrightarrow & \frac{-517x}{ \color{red}{-517} }& = & \frac{-658}{-517} \\\Leftrightarrow & x = \frac{14}{11} & & \\ & V = \left\{ \frac{14}{11} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{5}{4}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{165}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & 165x+792 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{-165x} & = & \color{red}{165x} +792 \color{blue}{-165x} \color{blue}{+48} \\\Leftrightarrow & -121x& = & 840 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{840}{-121} \\\Leftrightarrow & x = \frac{-840}{121} & & \\ & V = \left\{ \frac{-840}{121} \right\} & \\\end{align}\)