Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 28x+588 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-28x} & = & \color{red}{28x} +588 \color{blue}{-28x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 564 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{564}{-7} \\\Leftrightarrow & x = \frac{-564}{7} & & \\ & V = \left\{ \frac{-564}{7} \right\} & \\\end{align}\)
2. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{264}{ \color{blue}{330} }x+\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & 264x+660 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{-264x} & = & \color{red}{264x} +660 \color{blue}{-264x} \color{blue}{-90} \\\Leftrightarrow & -209x& = & 570 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{570}{-209} \\\Leftrightarrow & x = \frac{-30}{11} & & \\ & V = \left\{ \frac{-30}{11} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & -546x+780 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{+546x} & = & \color{red}{-546x} +780 \color{blue}{+546x} \color{blue}{-150} \\\Leftrightarrow & 611x& = & 630 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{630}{611} \\\Leftrightarrow & x = \frac{630}{611} & & \\ & V = \left\{ \frac{630}{611} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{5}{6}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 25x+30 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-25x} & = & \color{red}{25x} +30 \color{blue}{-25x} \color{blue}{-25} \\\Leftrightarrow & -19x& = & 5 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{5}{-19} \\\Leftrightarrow & x = \frac{-5}{19} & & \\ & V = \left\{ \frac{-5}{19} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{7}{2}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{315}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 315x+450 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-315x} & = & \color{red}{315x} +450 \color{blue}{-315x} \color{blue}{-40} \\\Leftrightarrow & -297x& = & 410 \\\Leftrightarrow & \frac{-297x}{ \color{red}{-297} }& = & \frac{410}{-297} \\\Leftrightarrow & x = \frac{-410}{297} & & \\ & V = \left\{ \frac{-410}{297} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & 6x-240 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} -240 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & 9x& = & -244 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{-244}{9} \\\Leftrightarrow & x = \frac{-244}{9} & & \\ & V = \left\{ \frac{-244}{9} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{3}{4}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 63x-672 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-63x} & = & \color{red}{63x} -672 \color{blue}{-63x} \color{blue}{+36} \\\Leftrightarrow & -35x& = & -636 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-636}{-35} \\\Leftrightarrow & x = \frac{636}{35} & & \\ & V = \left\{ \frac{636}{35} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{14}& = & \frac{-5}{6}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+45& = & -175x+1050 \\\Leftrightarrow & 42x \color{red}{+45} \color{blue}{-45} \color{blue}{+175x} & = & \color{red}{-175x} +1050 \color{blue}{+175x} \color{blue}{-45} \\\Leftrightarrow & 217x& = & 1005 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{1005}{217} \\\Leftrightarrow & x = \frac{1005}{217} & & \\ & V = \left\{ \frac{1005}{217} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{2}{3}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{104}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & 104x-312 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{-104x} & = & \color{red}{104x} -312 \color{blue}{-104x} \color{blue}{+36} \\\Leftrightarrow & -65x& = & -276 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-276}{-65} \\\Leftrightarrow & x = \frac{276}{65} & & \\ & V = \left\{ \frac{276}{65} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{7}{4}x-6 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{63}{ \color{blue}{36} }x-\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & 63x-216 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-63x} & = & \color{red}{63x} -216 \color{blue}{-63x} \color{blue}{+8} \\\Leftrightarrow & -51x& = & -208 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{-208}{-51} \\\Leftrightarrow & x = \frac{208}{51} & & \\ & V = \left\{ \frac{208}{51} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & 16x+288 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{-16x} & = & \color{red}{16x} +288 \color{blue}{-16x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & 279 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{279}{-4} \\\Leftrightarrow & x = \frac{-279}{4} & & \\ & V = \left\{ \frac{-279}{4} \right\} & \\\end{align}\)
12. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 60 }{ \color{blue}{195} })& = & (\frac{-130}{ \color{blue}{195} }x-\frac{1170}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-60& = & -130x-1170 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{+130x} & = & \color{red}{-130x} -1170 \color{blue}{+130x} \color{blue}{+60} \\\Leftrightarrow & 169x& = & -1110 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-1110}{169} \\\Leftrightarrow & x = \frac{-1110}{169} & & \\ & V = \left\{ \frac{-1110}{169} \right\} & \\\end{align}\)