Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x-\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-25& = & 16x-80 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{-16x} & = & \color{red}{16x} -80 \color{blue}{-16x} \color{blue}{+25} \\\Leftrightarrow & 4x& = & -55 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-55}{4} \\\Leftrightarrow & x = \frac{-55}{4} & & \\ & V = \left\{ \frac{-55}{4} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 50 }{ \color{blue}{60} })& = & (\frac{-24}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+50& = & -24x+240 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{+24x} & = & \color{red}{-24x} +240 \color{blue}{+24x} \color{blue}{-50} \\\Leftrightarrow & 39x& = & 190 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{190}{39} \\\Leftrightarrow & x = \frac{190}{39} & & \\ & V = \left\{ \frac{190}{39} \right\} & \\\end{align}\)
3. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 75 }{ \color{blue}{195} })& = & (\frac{-520}{ \color{blue}{195} }x-\frac{1170}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-75& = & -520x-1170 \\\Leftrightarrow & 39x \color{red}{-75} \color{blue}{+75} \color{blue}{+520x} & = & \color{red}{-520x} -1170 \color{blue}{+520x} \color{blue}{+75} \\\Leftrightarrow & 559x& = & -1095 \\\Leftrightarrow & \frac{559x}{ \color{red}{559} }& = & \frac{-1095}{559} \\\Leftrightarrow & x = \frac{-1095}{559} & & \\ & V = \left\{ \frac{-1095}{559} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{11}& = & \frac{2}{3}x-1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 36 }{ \color{blue}{132} })& = & (\frac{88}{ \color{blue}{132} }x-\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+36& = & 88x-132 \\\Leftrightarrow & 33x \color{red}{+36} \color{blue}{-36} \color{blue}{-88x} & = & \color{red}{88x} -132 \color{blue}{-88x} \color{blue}{-36} \\\Leftrightarrow & -55x& = & -168 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-168}{-55} \\\Leftrightarrow & x = \frac{168}{55} & & \\ & V = \left\{ \frac{168}{55} \right\} & \\\end{align}\)
5. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{1456}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 91x-1456 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-91x} & = & \color{red}{91x} -1456 \color{blue}{-91x} \color{blue}{+42} \\\Leftrightarrow & -65x& = & -1414 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-1414}{-65} \\\Leftrightarrow & x = \frac{1414}{65} & & \\ & V = \left\{ \frac{1414}{65} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{42}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & 42x-84 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{-42x} & = & \color{red}{42x} -84 \color{blue}{-42x} \color{blue}{+35} \\\Leftrightarrow & -30x& = & -49 \\\Leftrightarrow & \frac{-30x}{ \color{red}{-30} }& = & \frac{-49}{-30} \\\Leftrightarrow & x = \frac{49}{30} & & \\ & V = \left\{ \frac{49}{30} \right\} & \\\end{align}\)
7. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-5}{6}x-8 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 168 }{ \color{blue}{546} })& = & (\frac{-455}{ \color{blue}{546} }x-\frac{4368}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-168& = & -455x-4368 \\\Leftrightarrow & 78x \color{red}{-168} \color{blue}{+168} \color{blue}{+455x} & = & \color{red}{-455x} -4368 \color{blue}{+455x} \color{blue}{+168} \\\Leftrightarrow & 533x& = & -4200 \\\Leftrightarrow & \frac{533x}{ \color{red}{533} }& = & \frac{-4200}{533} \\\Leftrightarrow & x = \frac{-4200}{533} & & \\ & V = \left\{ \frac{-4200}{533} \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{42}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 7x+42 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-7x} & = & \color{red}{7x} +42 \color{blue}{-7x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 46 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{46}{-5} \\\Leftrightarrow & x = \frac{-46}{5} & & \\ & V = \left\{ \frac{-46}{5} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{4}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 30 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+30& = & 56x+210 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-56x} & = & \color{red}{56x} +210 \color{blue}{-56x} \color{blue}{-30} \\\Leftrightarrow & -35x& = & 180 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{180}{-35} \\\Leftrightarrow & x = \frac{-36}{7} & & \\ & V = \left\{ \frac{-36}{7} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+8& = & 15x+150 \\\Leftrightarrow & 10x \color{red}{+8} \color{blue}{-8} \color{blue}{-15x} & = & \color{red}{15x} +150 \color{blue}{-15x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & 142 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{142}{-5} \\\Leftrightarrow & x = \frac{-142}{5} & & \\ & V = \left\{ \frac{-142}{5} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{4}{5}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 24x-30 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} -30 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -19x& = & -39 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-39}{-19} \\\Leftrightarrow & x = \frac{39}{19} & & \\ & V = \left\{ \frac{39}{19} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{13}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-150& = & -546x-3120 \\\Leftrightarrow & 65x \color{red}{-150} \color{blue}{+150} \color{blue}{+546x} & = & \color{red}{-546x} -3120 \color{blue}{+546x} \color{blue}{+150} \\\Leftrightarrow & 611x& = & -2970 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{-2970}{611} \\\Leftrightarrow & x = \frac{-2970}{611} & & \\ & V = \left\{ \frac{-2970}{611} \right\} & \\\end{align}\)