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. \(\frac{x}{4}+\frac{2}{13}=\frac{7}{3}x+1\)
2. \(\frac{x}{7}+\frac{5}{9}=\frac{-5}{6}x+6\)
3. \(\frac{x}{3}-\frac{3}{7}=\frac{1}{4}x+5\)
4. \(\frac{x}{7}-\frac{3}{7}=\frac{4}{3}x-5\)
5. \(\frac{x}{7}-\frac{4}{7}=\frac{1}{4}x-5\)
6. \(\frac{x}{3}+\frac{2}{13}=\frac{1}{2}x-6\)
7. \(\frac{x}{2}-\frac{3}{7}=\frac{1}{3}x+1\)
8. \(\frac{x}{6}+\frac{3}{14}=\frac{6}{5}x-7\)
9. \(\frac{x}{3}+\frac{4}{7}=\frac{5}{4}x+1\)
10. \(\frac{x}{4}-\frac{2}{11}=\frac{7}{5}x-2\)
11. \(\frac{x}{7}-\frac{3}{16}=\frac{-5}{3}x-3\)
12. \(\frac{x}{7}+\frac{2}{13}=\frac{1}{3}x+5\)

---

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{364}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+24& = & 364x+156 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-364x} & = & \color{red}{364x} +156 \color{blue}{-364x} \color{blue}{-24} \\\Leftrightarrow & -325x& = & 132 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{132}{-325} \\\Leftrightarrow & x = \frac{-132}{325} & & \\ & V = \left\{ \frac{-132}{325} \right\} & \\\end{align}\)
2. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 70 }{ \color{blue}{126} })& = & (\frac{-105}{ \color{blue}{126} }x+\frac{756}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+70& = & -105x+756 \\\Leftrightarrow & 18x \color{red}{+70} \color{blue}{-70} \color{blue}{+105x} & = & \color{red}{-105x} +756 \color{blue}{+105x} \color{blue}{-70} \\\Leftrightarrow & 123x& = & 686 \\\Leftrightarrow & \frac{123x}{ \color{red}{123} }& = & \frac{686}{123} \\\Leftrightarrow & x = \frac{686}{123} & & \\ & V = \left\{ \frac{686}{123} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 21x+420 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-21x} & = & \color{red}{21x} +420 \color{blue}{-21x} \color{blue}{+36} \\\Leftrightarrow & 7x& = & 456 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{456}{7} \\\Leftrightarrow & x = \frac{456}{7} & & \\ & V = \left\{ \frac{456}{7} \right\} & \\\end{align}\)
4. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 9 }{ \color{blue}{21} })& = & (\frac{28}{ \color{blue}{21} }x-\frac{105}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-9& = & 28x-105 \\\Leftrightarrow & 3x \color{red}{-9} \color{blue}{+9} \color{blue}{-28x} & = & \color{red}{28x} -105 \color{blue}{-28x} \color{blue}{+9} \\\Leftrightarrow & -25x& = & -96 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-96}{-25} \\\Leftrightarrow & x = \frac{96}{25} & & \\ & V = \left\{ \frac{96}{25} \right\} & \\\end{align}\)
5. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{4}x-5 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 16 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x-\frac{140}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-16& = & 7x-140 \\\Leftrightarrow & 4x \color{red}{-16} \color{blue}{+16} \color{blue}{-7x} & = & \color{red}{7x} -140 \color{blue}{-7x} \color{blue}{+16} \\\Leftrightarrow & -3x& = & -124 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-124}{-3} \\\Leftrightarrow & x = \frac{124}{3} & & \\ & V = \left\{ \frac{124}{3} \right\} & \\\end{align}\)
6. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 39x-468 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-39x} & = & \color{red}{39x} -468 \color{blue}{-39x} \color{blue}{-12} \\\Leftrightarrow & -13x& = & -480 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-480}{-13} \\\Leftrightarrow & x = \frac{480}{13} & & \\ & V = \left\{ \frac{480}{13} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 14x+42 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-14x} & = & \color{red}{14x} +42 \color{blue}{-14x} \color{blue}{+18} \\\Leftrightarrow & 7x& = & 60 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{60}{7} \\\Leftrightarrow & x = \frac{60}{7} & & \\ & V = \left\{ \frac{60}{7} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & 252x-1470 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{-252x} & = & \color{red}{252x} -1470 \color{blue}{-252x} \color{blue}{-45} \\\Leftrightarrow & -217x& = & -1515 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-1515}{-217} \\\Leftrightarrow & x = \frac{1515}{217} & & \\ & V = \left\{ \frac{1515}{217} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{5}{4}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & 105x+84 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{-105x} & = & \color{red}{105x} +84 \color{blue}{-105x} \color{blue}{-48} \\\Leftrightarrow & -77x& = & 36 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{36}{-77} \\\Leftrightarrow & x = \frac{-36}{77} & & \\ & V = \left\{ \frac{-36}{77} \right\} & \\\end{align}\)
10. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{7}{5}x-2 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{308}{ \color{blue}{220} }x-\frac{440}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-40& = & 308x-440 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-308x} & = & \color{red}{308x} -440 \color{blue}{-308x} \color{blue}{+40} \\\Leftrightarrow & -253x& = & -400 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{-400}{-253} \\\Leftrightarrow & x = \frac{400}{253} & & \\ & V = \left\{ \frac{400}{253} \right\} & \\\end{align}\)
11. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 63 }{ \color{blue}{336} })& = & (\frac{-560}{ \color{blue}{336} }x-\frac{1008}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-63& = & -560x-1008 \\\Leftrightarrow & 48x \color{red}{-63} \color{blue}{+63} \color{blue}{+560x} & = & \color{red}{-560x} -1008 \color{blue}{+560x} \color{blue}{+63} \\\Leftrightarrow & 608x& = & -945 \\\Leftrightarrow & \frac{608x}{ \color{red}{608} }& = & \frac{-945}{608} \\\Leftrightarrow & x = \frac{-945}{608} & & \\ & V = \left\{ \frac{-945}{608} \right\} & \\\end{align}\)
12. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 42 }{ \color{blue}{273} })& = & (\frac{91}{ \color{blue}{273} }x+\frac{1365}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+42& = & 91x+1365 \\\Leftrightarrow & 39x \color{red}{+42} \color{blue}{-42} \color{blue}{-91x} & = & \color{red}{91x} +1365 \color{blue}{-91x} \color{blue}{-42} \\\Leftrightarrow & -52x& = & 1323 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{1323}{-52} \\\Leftrightarrow & x = \frac{-1323}{52} & & \\ & V = \left\{ \frac{-1323}{52} \right\} & \\\end{align}\)