Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x+\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -96x+720 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+96x} & = & \color{red}{-96x} +720 \color{blue}{+96x} \color{blue}{-75} \\\Leftrightarrow & 136x& = & 645 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{645}{136} \\\Leftrightarrow & x = \frac{645}{136} & & \\ & V = \left\{ \frac{645}{136} \right\} & \\\end{align}\)
2. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-8}{3}x-3 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 63 }{ \color{blue}{273} })& = & (\frac{-728}{ \color{blue}{273} }x-\frac{819}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-63& = & -728x-819 \\\Leftrightarrow & 39x \color{red}{-63} \color{blue}{+63} \color{blue}{+728x} & = & \color{red}{-728x} -819 \color{blue}{+728x} \color{blue}{+63} \\\Leftrightarrow & 767x& = & -756 \\\Leftrightarrow & \frac{767x}{ \color{red}{767} }& = & \frac{-756}{767} \\\Leftrightarrow & x = \frac{-756}{767} & & \\ & V = \left\{ \frac{-756}{767} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & -40x+420 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+40x} & = & \color{red}{-40x} +420 \color{blue}{+40x} \color{blue}{+8} \\\Leftrightarrow & 55x& = & 428 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{428}{55} \\\Leftrightarrow & x = \frac{428}{55} & & \\ & V = \left\{ \frac{428}{55} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x+630 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} +630 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & 670 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{670}{-27} \\\Leftrightarrow & x = \frac{-670}{27} & & \\ & V = \left\{ \frac{-670}{27} \right\} & \\\end{align}\)
5. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{7}{3}x+6 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 105 }{ \color{blue}{336} })& = & (\frac{784}{ \color{blue}{336} }x+\frac{2016}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+105& = & 784x+2016 \\\Leftrightarrow & 48x \color{red}{+105} \color{blue}{-105} \color{blue}{-784x} & = & \color{red}{784x} +2016 \color{blue}{-784x} \color{blue}{-105} \\\Leftrightarrow & -736x& = & 1911 \\\Leftrightarrow & \frac{-736x}{ \color{red}{-736} }& = & \frac{1911}{-736} \\\Leftrightarrow & x = \frac{-1911}{736} & & \\ & V = \left\{ \frac{-1911}{736} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x-90 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} -90 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & -130 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-130}{-3} \\\Leftrightarrow & x = \frac{130}{3} & & \\ & V = \left\{ \frac{130}{3} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+25& = & 35x+70 \\\Leftrightarrow & 14x \color{red}{+25} \color{blue}{-25} \color{blue}{-35x} & = & \color{red}{35x} +70 \color{blue}{-35x} \color{blue}{-25} \\\Leftrightarrow & -21x& = & 45 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{45}{-21} \\\Leftrightarrow & x = \frac{-15}{7} & & \\ & V = \left\{ \frac{-15}{7} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & 175x+420 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{-175x} & = & \color{red}{175x} +420 \color{blue}{-175x} \color{blue}{-120} \\\Leftrightarrow & -133x& = & 300 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{300}{-133} \\\Leftrightarrow & x = \frac{-300}{133} & & \\ & V = \left\{ \frac{-300}{133} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{5}{3}x+6 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{260}{ \color{blue}{156} }x+\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & 260x+936 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-260x} & = & \color{red}{260x} +936 \color{blue}{-260x} \color{blue}{+24} \\\Leftrightarrow & -221x& = & 960 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{960}{-221} \\\Leftrightarrow & x = \frac{-960}{221} & & \\ & V = \left\{ \frac{-960}{221} \right\} & \\\end{align}\)
10. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x-\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & 48x-1680 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-48x} & = & \color{red}{48x} -1680 \color{blue}{-48x} \color{blue}{-45} \\\Leftrightarrow & -8x& = & -1725 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-1725}{-8} \\\Leftrightarrow & x = \frac{1725}{8} & & \\ & V = \left\{ \frac{1725}{8} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{3}{4}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 63x+336 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-63x} & = & \color{red}{63x} +336 \color{blue}{-63x} \color{blue}{+36} \\\Leftrightarrow & -35x& = & 372 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{372}{-35} \\\Leftrightarrow & x = \frac{-372}{35} & & \\ & V = \left\{ \frac{-372}{35} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x-\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & -80x-192 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{+80x} & = & \color{red}{-80x} -192 \color{blue}{+80x} \color{blue}{-9} \\\Leftrightarrow & 92x& = & -201 \\\Leftrightarrow & \frac{92x}{ \color{red}{92} }& = & \frac{-201}{92} \\\Leftrightarrow & x = \frac{-201}{92} & & \\ & V = \left\{ \frac{-201}{92} \right\} & \\\end{align}\)