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

---

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{7}{2}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 28 }{ \color{blue}{210} })& = & (\frac{735}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+28& = & 735x+630 \\\Leftrightarrow & 30x \color{red}{+28} \color{blue}{-28} \color{blue}{-735x} & = & \color{red}{735x} +630 \color{blue}{-735x} \color{blue}{-28} \\\Leftrightarrow & -705x& = & 602 \\\Leftrightarrow & \frac{-705x}{ \color{red}{-705} }& = & \frac{602}{-705} \\\Leftrightarrow & x = \frac{-602}{705} & & \\ & V = \left\{ \frac{-602}{705} \right\} & \\\end{align}\)
2. \(\text{120 is het kleinste gemene veelvoud van 3, 8 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{120.} (\frac{40x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{144}{ \color{blue}{120} }x-\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 40x+45& = & 144x-480 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-144x} & = & \color{red}{144x} -480 \color{blue}{-144x} \color{blue}{-45} \\\Leftrightarrow & -104x& = & -525 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{-525}{-104} \\\Leftrightarrow & x = \frac{525}{104} & & \\ & V = \left\{ \frac{525}{104} \right\} & \\\end{align}\)
3. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x+\frac{1040}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+20& = & 65x+1040 \\\Leftrightarrow & 26x \color{red}{+20} \color{blue}{-20} \color{blue}{-65x} & = & \color{red}{65x} +1040 \color{blue}{-65x} \color{blue}{-20} \\\Leftrightarrow & -39x& = & 1020 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{1020}{-39} \\\Leftrightarrow & x = \frac{-340}{13} & & \\ & V = \left\{ \frac{-340}{13} \right\} & \\\end{align}\)
4. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 12 }{ \color{blue}{21} })& = & (\frac{28}{ \color{blue}{21} }x+\frac{42}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-12& = & 28x+42 \\\Leftrightarrow & 3x \color{red}{-12} \color{blue}{+12} \color{blue}{-28x} & = & \color{red}{28x} +42 \color{blue}{-28x} \color{blue}{+12} \\\Leftrightarrow & -25x& = & 54 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{54}{-25} \\\Leftrightarrow & x = \frac{-54}{25} & & \\ & V = \left\{ \frac{-54}{25} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{7}{6}x+4 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{385}{ \color{blue}{330} }x+\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+60& = & 385x+1320 \\\Leftrightarrow & 66x \color{red}{+60} \color{blue}{-60} \color{blue}{-385x} & = & \color{red}{385x} +1320 \color{blue}{-385x} \color{blue}{-60} \\\Leftrightarrow & -319x& = & 1260 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{1260}{-319} \\\Leftrightarrow & x = \frac{-1260}{319} & & \\ & V = \left\{ \frac{-1260}{319} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -56x-252 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+56x} & = & \color{red}{-56x} -252 \color{blue}{+56x} \color{blue}{-48} \\\Leftrightarrow & 77x& = & -300 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-300}{77} \\\Leftrightarrow & x = \frac{-300}{77} & & \\ & V = \left\{ \frac{-300}{77} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -24x-180 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+24x} & = & \color{red}{-24x} -180 \color{blue}{+24x} \color{blue}{+4} \\\Leftrightarrow & 29x& = & -176 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-176}{29} \\\Leftrightarrow & x = \frac{-176}{29} & & \\ & V = \left\{ \frac{-176}{29} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -294x+840 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+294x} & = & \color{red}{-294x} +840 \color{blue}{+294x} \color{blue}{-60} \\\Leftrightarrow & 329x& = & 780 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{780}{329} \\\Leftrightarrow & x = \frac{780}{329} & & \\ & V = \left\{ \frac{780}{329} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{1}{3}x-8 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x-\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 22x-528 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-22x} & = & \color{red}{22x} -528 \color{blue}{-22x} \color{blue}{+12} \\\Leftrightarrow & 11x& = & -516 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-516}{11} \\\Leftrightarrow & x = \frac{-516}{11} & & \\ & V = \left\{ \frac{-516}{11} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x+\frac{252}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-16& = & -27x+252 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{+27x} & = & \color{red}{-27x} +252 \color{blue}{+27x} \color{blue}{+16} \\\Leftrightarrow & 39x& = & 268 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{268}{39} \\\Leftrightarrow & x = \frac{268}{39} & & \\ & V = \left\{ \frac{268}{39} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{5}{2}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{210}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & 210x-336 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{-210x} & = & \color{red}{210x} -336 \color{blue}{-210x} \color{blue}{+35} \\\Leftrightarrow & -198x& = & -301 \\\Leftrightarrow & \frac{-198x}{ \color{red}{-198} }& = & \frac{-301}{-198} \\\Leftrightarrow & x = \frac{301}{198} & & \\ & V = \left\{ \frac{301}{198} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & -546x-3120 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{+546x} & = & \color{red}{-546x} -3120 \color{blue}{+546x} \color{blue}{+60} \\\Leftrightarrow & 611x& = & -3060 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{-3060}{611} \\\Leftrightarrow & x = \frac{-3060}{611} & & \\ & V = \left\{ \frac{-3060}{611} \right\} & \\\end{align}\)