Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{3}{5}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{36}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 36x+180 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-36x} & = & \color{red}{36x} +180 \color{blue}{-36x} \color{blue}{-8} \\\Leftrightarrow & -21x& = & 172 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{172}{-21} \\\Leftrightarrow & x = \frac{-172}{21} & & \\ & V = \left\{ \frac{-172}{21} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-8}{3}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-208}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & -208x-312 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{+208x} & = & \color{red}{-208x} -312 \color{blue}{+208x} \color{blue}{+18} \\\Leftrightarrow & 247x& = & -294 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-294}{247} \\\Leftrightarrow & x = \frac{-294}{247} & & \\ & V = \left\{ \frac{-294}{247} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 468x-1560 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-468x} & = & \color{red}{468x} -1560 \color{blue}{-468x} \color{blue}{-60} \\\Leftrightarrow & -403x& = & -1620 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-1620}{-403} \\\Leftrightarrow & x = \frac{1620}{403} & & \\ & V = \left\{ \frac{1620}{403} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x+90 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} +90 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & 81 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{81}{-1} \\\Leftrightarrow & x = -81 & & \\ & V = \left\{ -81 \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+9& = & 10x+30 \\\Leftrightarrow & 15x \color{red}{+9} \color{blue}{-9} \color{blue}{-10x} & = & \color{red}{10x} +30 \color{blue}{-10x} \color{blue}{-9} \\\Leftrightarrow & 5x& = & 21 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{21}{5} \\\Leftrightarrow & x = \frac{21}{5} & & \\ & V = \left\{ \frac{21}{5} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{9}& = & \frac{2}{5}x-5 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{36}{ \color{blue}{90} }x-\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+50& = & 36x-450 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{-36x} & = & \color{red}{36x} -450 \color{blue}{-36x} \color{blue}{-50} \\\Leftrightarrow & -21x& = & -500 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-500}{-21} \\\Leftrightarrow & x = \frac{500}{21} & & \\ & V = \left\{ \frac{500}{21} \right\} & \\\end{align}\)
7. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 78x+780 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-78x} & = & \color{red}{78x} +780 \color{blue}{-78x} \color{blue}{-90} \\\Leftrightarrow & -13x& = & 690 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{690}{-13} \\\Leftrightarrow & x = \frac{-690}{13} & & \\ & V = \left\{ \frac{-690}{13} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & 252x+1680 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{-252x} & = & \color{red}{252x} +1680 \color{blue}{-252x} \color{blue}{-45} \\\Leftrightarrow & -217x& = & 1635 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{1635}{-217} \\\Leftrightarrow & x = \frac{-1635}{217} & & \\ & V = \left\{ \frac{-1635}{217} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 14x+294 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-14x} & = & \color{red}{14x} +294 \color{blue}{-14x} \color{blue}{+24} \\\Leftrightarrow & 7x& = & 318 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{318}{7} \\\Leftrightarrow & x = \frac{318}{7} & & \\ & V = \left\{ \frac{318}{7} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{7}{2}x+3 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{280}{ \color{blue}{80} }x+\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 280x+240 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-280x} & = & \color{red}{280x} +240 \color{blue}{-280x} \color{blue}{-25} \\\Leftrightarrow & -264x& = & 215 \\\Leftrightarrow & \frac{-264x}{ \color{red}{-264} }& = & \frac{215}{-264} \\\Leftrightarrow & x = \frac{-215}{264} & & \\ & V = \left\{ \frac{-215}{264} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-8}{3}x-4 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-352}{ \color{blue}{132} }x-\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -352x-528 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+352x} & = & \color{red}{-352x} -528 \color{blue}{+352x} \color{blue}{-24} \\\Leftrightarrow & 385x& = & -552 \\\Leftrightarrow & \frac{385x}{ \color{red}{385} }& = & \frac{-552}{385} \\\Leftrightarrow & x = \frac{-552}{385} & & \\ & V = \left\{ \frac{-552}{385} \right\} & \\\end{align}\)
12. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{-72}{ \color{blue}{180} }x+\frac{360}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+80& = & -72x+360 \\\Leftrightarrow & 45x \color{red}{+80} \color{blue}{-80} \color{blue}{+72x} & = & \color{red}{-72x} +360 \color{blue}{+72x} \color{blue}{-80} \\\Leftrightarrow & 117x& = & 280 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{280}{117} \\\Leftrightarrow & x = \frac{280}{117} & & \\ & V = \left\{ \frac{280}{117} \right\} & \\\end{align}\)