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

---

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+36& = & 208x-312 \\\Leftrightarrow & 39x \color{red}{+36} \color{blue}{-36} \color{blue}{-208x} & = & \color{red}{208x} -312 \color{blue}{-208x} \color{blue}{-36} \\\Leftrightarrow & -169x& = & -348 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-348}{-169} \\\Leftrightarrow & x = \frac{348}{169} & & \\ & V = \left\{ \frac{348}{169} \right\} & \\\end{align}\)
2. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{9}& = & \frac{1}{4}x+6 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }- \frac{ 100 }{ \color{blue}{180} })& = & (\frac{45}{ \color{blue}{180} }x+\frac{1080}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x-100& = & 45x+1080 \\\Leftrightarrow & 36x \color{red}{-100} \color{blue}{+100} \color{blue}{-45x} & = & \color{red}{45x} +1080 \color{blue}{-45x} \color{blue}{+100} \\\Leftrightarrow & -9x& = & 1180 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{1180}{-9} \\\Leftrightarrow & x = \frac{-1180}{9} & & \\ & V = \left\{ \frac{-1180}{9} \right\} & \\\end{align}\)
3. \(\text{120 is het kleinste gemene veelvoud van 3, 8 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{120.} (\frac{40x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x-\frac{120}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 40x+45& = & -168x-120 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{+168x} & = & \color{red}{-168x} -120 \color{blue}{+168x} \color{blue}{-45} \\\Leftrightarrow & 208x& = & -165 \\\Leftrightarrow & \frac{208x}{ \color{red}{208} }& = & \frac{-165}{208} \\\Leftrightarrow & x = \frac{-165}{208} & & \\ & V = \left\{ \frac{-165}{208} \right\} & \\\end{align}\)
4. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{3}{4}x+8 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{30}{ \color{blue}{40} }x+\frac{320}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x+25& = & 30x+320 \\\Leftrightarrow & 8x \color{red}{+25} \color{blue}{-25} \color{blue}{-30x} & = & \color{red}{30x} +320 \color{blue}{-30x} \color{blue}{-25} \\\Leftrightarrow & -22x& = & 295 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{295}{-22} \\\Leftrightarrow & x = \frac{-295}{22} & & \\ & V = \left\{ \frac{-295}{22} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -12x+90 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+12x} & = & \color{red}{-12x} +90 \color{blue}{+12x} \color{blue}{-9} \\\Leftrightarrow & 17x& = & 81 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{81}{17} \\\Leftrightarrow & x = \frac{81}{17} & & \\ & V = \left\{ \frac{81}{17} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 21x+336 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-21x} & = & \color{red}{21x} +336 \color{blue}{-21x} \color{blue}{+36} \\\Leftrightarrow & 7x& = & 372 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{372}{7} \\\Leftrightarrow & x = \frac{372}{7} & & \\ & V = \left\{ \frac{372}{7} \right\} & \\\end{align}\)
7. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 9 }{ \color{blue}{21} })& = & (\frac{14}{ \color{blue}{21} }x+\frac{42}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+9& = & 14x+42 \\\Leftrightarrow & 3x \color{red}{+9} \color{blue}{-9} \color{blue}{-14x} & = & \color{red}{14x} +42 \color{blue}{-14x} \color{blue}{-9} \\\Leftrightarrow & -11x& = & 33 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{33}{-11} \\\Leftrightarrow & x = -3 & & \\ & V = \left\{ -3 \right\} & \\\end{align}\)
8. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }+ \frac{ 2 }{ \color{blue}{15} })& = & (\frac{5}{ \color{blue}{15} }x-\frac{105}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x+2& = & 5x-105 \\\Leftrightarrow & 3x \color{red}{+2} \color{blue}{-2} \color{blue}{-5x} & = & \color{red}{5x} -105 \color{blue}{-5x} \color{blue}{-2} \\\Leftrightarrow & -2x& = & -107 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-107}{-2} \\\Leftrightarrow & x = \frac{107}{2} & & \\ & V = \left\{ \frac{107}{2} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -294x-1260 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+294x} & = & \color{red}{-294x} -1260 \color{blue}{+294x} \color{blue}{-120} \\\Leftrightarrow & 329x& = & -1380 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-1380}{329} \\\Leftrightarrow & x = \frac{-1380}{329} & & \\ & V = \left\{ \frac{-1380}{329} \right\} & \\\end{align}\)
10. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-5}{6}x+1 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 126 }{ \color{blue}{546} })& = & (\frac{-455}{ \color{blue}{546} }x+\frac{546}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-126& = & -455x+546 \\\Leftrightarrow & 78x \color{red}{-126} \color{blue}{+126} \color{blue}{+455x} & = & \color{red}{-455x} +546 \color{blue}{+455x} \color{blue}{+126} \\\Leftrightarrow & 533x& = & 672 \\\Leftrightarrow & \frac{533x}{ \color{red}{533} }& = & \frac{672}{533} \\\Leftrightarrow & x = \frac{672}{533} & & \\ & V = \left\{ \frac{672}{533} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{-8}{3}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & -176x-264 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{+176x} & = & \color{red}{-176x} -264 \color{blue}{+176x} \color{blue}{-12} \\\Leftrightarrow & 209x& = & -276 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-276}{209} \\\Leftrightarrow & x = \frac{-276}{209} & & \\ & V = \left\{ \frac{-276}{209} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -126x+270 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+126x} & = & \color{red}{-126x} +270 \color{blue}{+126x} \color{blue}{-40} \\\Leftrightarrow & 141x& = & 230 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{230}{141} \\\Leftrightarrow & x = \frac{230}{141} & & \\ & V = \left\{ \frac{230}{141} \right\} & \\\end{align}\)