Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{13}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 70 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{1456}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-70& = & 91x+1456 \\\Leftrightarrow & 26x \color{red}{-70} \color{blue}{+70} \color{blue}{-91x} & = & \color{red}{91x} +1456 \color{blue}{-91x} \color{blue}{+70} \\\Leftrightarrow & -65x& = & 1526 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{1526}{-65} \\\Leftrightarrow & x = \frac{-1526}{65} & & \\ & V = \left\{ \frac{-1526}{65} \right\} & \\\end{align}\)
2. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 105 }{ \color{blue}{336} })& = & (\frac{-224}{ \color{blue}{336} }x-\frac{672}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-105& = & -224x-672 \\\Leftrightarrow & 48x \color{red}{-105} \color{blue}{+105} \color{blue}{+224x} & = & \color{red}{-224x} -672 \color{blue}{+224x} \color{blue}{+105} \\\Leftrightarrow & 272x& = & -567 \\\Leftrightarrow & \frac{272x}{ \color{red}{272} }& = & \frac{-567}{272} \\\Leftrightarrow & x = \frac{-567}{272} & & \\ & V = \left\{ \frac{-567}{272} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 30 }{ \color{blue}{165} })& = & (\frac{198}{ \color{blue}{165} }x-\frac{660}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-30& = & 198x-660 \\\Leftrightarrow & 55x \color{red}{-30} \color{blue}{+30} \color{blue}{-198x} & = & \color{red}{198x} -660 \color{blue}{-198x} \color{blue}{+30} \\\Leftrightarrow & -143x& = & -630 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-630}{-143} \\\Leftrightarrow & x = \frac{630}{143} & & \\ & V = \left\{ \frac{630}{143} \right\} & \\\end{align}\)
4. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 60 }{ \color{blue}{165} })& = & (\frac{-275}{ \color{blue}{165} }x+\frac{165}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-60& = & -275x+165 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{+275x} & = & \color{red}{-275x} +165 \color{blue}{+275x} \color{blue}{+60} \\\Leftrightarrow & 308x& = & 225 \\\Leftrightarrow & \frac{308x}{ \color{red}{308} }& = & \frac{225}{308} \\\Leftrightarrow & x = \frac{225}{308} & & \\ & V = \left\{ \frac{225}{308} \right\} & \\\end{align}\)
5. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{882}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 63x+882 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-63x} & = & \color{red}{63x} +882 \color{blue}{-63x} \color{blue}{-56} \\\Leftrightarrow & -45x& = & 826 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{826}{-45} \\\Leftrightarrow & x = \frac{-826}{45} & & \\ & V = \left\{ \frac{-826}{45} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{1}{3}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-48& = & 52x-312 \\\Leftrightarrow & 39x \color{red}{-48} \color{blue}{+48} \color{blue}{-52x} & = & \color{red}{52x} -312 \color{blue}{-52x} \color{blue}{+48} \\\Leftrightarrow & -13x& = & -264 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-264}{-13} \\\Leftrightarrow & x = \frac{264}{13} & & \\ & V = \left\{ \frac{264}{13} \right\} & \\\end{align}\)
7. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 40x-400 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-40x} & = & \color{red}{40x} -400 \color{blue}{-40x} \color{blue}{-25} \\\Leftrightarrow & -24x& = & -425 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-425}{-24} \\\Leftrightarrow & x = \frac{425}{24} & & \\ & V = \left\{ \frac{425}{24} \right\} & \\\end{align}\)
8. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{-2}{5}x-4 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{-44}{ \color{blue}{110} }x-\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+40& = & -44x-440 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{+44x} & = & \color{red}{-44x} -440 \color{blue}{+44x} \color{blue}{-40} \\\Leftrightarrow & 99x& = & -480 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{-480}{99} \\\Leftrightarrow & x = \frac{-160}{33} & & \\ & V = \left\{ \frac{-160}{33} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 15x+90 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} +90 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -9x& = & 98 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{98}{-9} \\\Leftrightarrow & x = \frac{-98}{9} & & \\ & V = \left\{ \frac{-98}{9} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{-4}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & -168x+840 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{+168x} & = & \color{red}{-168x} +840 \color{blue}{+168x} \color{blue}{-75} \\\Leftrightarrow & 203x& = & 765 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{765}{203} \\\Leftrightarrow & x = \frac{765}{203} & & \\ & V = \left\{ \frac{765}{203} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & -28x+168 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{+28x} & = & \color{red}{-28x} +168 \color{blue}{+28x} \color{blue}{+9} \\\Leftrightarrow & 49x& = & 177 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{177}{49} \\\Leftrightarrow & x = \frac{177}{49} & & \\ & V = \left\{ \frac{177}{49} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & -32x+288 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{+32x} & = & \color{red}{-32x} +288 \color{blue}{+32x} \color{blue}{+15} \\\Leftrightarrow & 56x& = & 303 \\\Leftrightarrow & \frac{56x}{ \color{red}{56} }& = & \frac{303}{56} \\\Leftrightarrow & x = \frac{303}{56} & & \\ & V = \left\{ \frac{303}{56} \right\} & \\\end{align}\)