Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}-\frac{2}{13}=\frac{-3}{4}x-1\)
2. \(\frac{x}{5}-\frac{4}{7}=\frac{2}{3}x+6\)
3. \(\frac{x}{7}+\frac{5}{12}=\frac{1}{2}x+7\)
4. \(\frac{x}{7}+\frac{3}{13}=\frac{7}{2}x-6\)
5. \(\frac{x}{2}-\frac{3}{16}=\frac{1}{5}x-2\)
6. \(\frac{x}{2}-\frac{3}{14}=\frac{7}{3}x+1\)
7. \(\frac{x}{2}+\frac{4}{7}=\frac{-5}{3}x-7\)
8. \(\frac{x}{5}-\frac{5}{12}=\frac{-5}{6}x-5\)
9. \(\frac{x}{6}+\frac{4}{9}=\frac{6}{5}x-3\)
10. \(\frac{x}{4}-\frac{4}{13}=\frac{1}{5}x+7\)
11. \(\frac{x}{4}+\frac{2}{15}=\frac{7}{3}x+1\)
12. \(\frac{x}{5}-\frac{4}{9}=\frac{1}{2}x+7\)

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

Verbetersleutel

1. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{-3}{4}x-1 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 56 }{ \color{blue}{364} })& = & (\frac{-273}{ \color{blue}{364} }x-\frac{364}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-56& = & -273x-364 \\\Leftrightarrow & 52x \color{red}{-56} \color{blue}{+56} \color{blue}{+273x} & = & \color{red}{-273x} -364 \color{blue}{+273x} \color{blue}{+56} \\\Leftrightarrow & 325x& = & -308 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{-308}{325} \\\Leftrightarrow & x = \frac{-308}{325} & & \\ & V = \left\{ \frac{-308}{325} \right\} & \\\end{align}\)
2. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{2}{3}x+6 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{70}{ \color{blue}{105} }x+\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-60& = & 70x+630 \\\Leftrightarrow & 21x \color{red}{-60} \color{blue}{+60} \color{blue}{-70x} & = & \color{red}{70x} +630 \color{blue}{-70x} \color{blue}{+60} \\\Leftrightarrow & -49x& = & 690 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{690}{-49} \\\Leftrightarrow & x = \frac{-690}{49} & & \\ & V = \left\{ \frac{-690}{49} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{12}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 35 }{ \color{blue}{84} })& = & (\frac{42}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+35& = & 42x+588 \\\Leftrightarrow & 12x \color{red}{+35} \color{blue}{-35} \color{blue}{-42x} & = & \color{red}{42x} +588 \color{blue}{-42x} \color{blue}{-35} \\\Leftrightarrow & -30x& = & 553 \\\Leftrightarrow & \frac{-30x}{ \color{red}{-30} }& = & \frac{553}{-30} \\\Leftrightarrow & x = \frac{-553}{30} & & \\ & V = \left\{ \frac{-553}{30} \right\} & \\\end{align}\)
4. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{7}{2}x-6 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 42 }{ \color{blue}{182} })& = & (\frac{637}{ \color{blue}{182} }x-\frac{1092}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+42& = & 637x-1092 \\\Leftrightarrow & 26x \color{red}{+42} \color{blue}{-42} \color{blue}{-637x} & = & \color{red}{637x} -1092 \color{blue}{-637x} \color{blue}{-42} \\\Leftrightarrow & -611x& = & -1134 \\\Leftrightarrow & \frac{-611x}{ \color{red}{-611} }& = & \frac{-1134}{-611} \\\Leftrightarrow & x = \frac{1134}{611} & & \\ & V = \left\{ \frac{1134}{611} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x-\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-15& = & 16x-160 \\\Leftrightarrow & 40x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} -160 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & 24x& = & -145 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{-145}{24} \\\Leftrightarrow & x = \frac{-145}{24} & & \\ & V = \left\{ \frac{-145}{24} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & 98x+42 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{-98x} & = & \color{red}{98x} +42 \color{blue}{-98x} \color{blue}{+9} \\\Leftrightarrow & -77x& = & 51 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{51}{-77} \\\Leftrightarrow & x = \frac{-51}{77} & & \\ & V = \left\{ \frac{-51}{77} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -70x-294 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+70x} & = & \color{red}{-70x} -294 \color{blue}{+70x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -318 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-318}{91} \\\Leftrightarrow & x = \frac{-318}{91} & & \\ & V = \left\{ \frac{-318}{91} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-50}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & -50x-300 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{+50x} & = & \color{red}{-50x} -300 \color{blue}{+50x} \color{blue}{+25} \\\Leftrightarrow & 62x& = & -275 \\\Leftrightarrow & \frac{62x}{ \color{red}{62} }& = & \frac{-275}{62} \\\Leftrightarrow & x = \frac{-275}{62} & & \\ & V = \left\{ \frac{-275}{62} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x-\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 108x-270 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-108x} & = & \color{red}{108x} -270 \color{blue}{-108x} \color{blue}{-40} \\\Leftrightarrow & -93x& = & -310 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{-310}{-93} \\\Leftrightarrow & x = \frac{10}{3} & & \\ & V = \left\{ \frac{10}{3} \right\} & \\\end{align}\)
10. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{52}{ \color{blue}{260} }x+\frac{1820}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-80& = & 52x+1820 \\\Leftrightarrow & 65x \color{red}{-80} \color{blue}{+80} \color{blue}{-52x} & = & \color{red}{52x} +1820 \color{blue}{-52x} \color{blue}{+80} \\\Leftrightarrow & 13x& = & 1900 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{1900}{13} \\\Leftrightarrow & x = \frac{1900}{13} & & \\ & V = \left\{ \frac{1900}{13} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 140x+60 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-140x} & = & \color{red}{140x} +60 \color{blue}{-140x} \color{blue}{-8} \\\Leftrightarrow & -125x& = & 52 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{52}{-125} \\\Leftrightarrow & x = \frac{-52}{125} & & \\ & V = \left\{ \frac{-52}{125} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x+630 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} +630 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & 670 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{670}{-27} \\\Leftrightarrow & x = \frac{-670}{27} & & \\ & V = \left\{ \frac{-670}{27} \right\} & \\\end{align}\)