Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 84 }{ \color{blue}{273} })& = & (\frac{-455}{ \color{blue}{273} }x+\frac{819}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-84& = & -455x+819 \\\Leftrightarrow & 39x \color{red}{-84} \color{blue}{+84} \color{blue}{+455x} & = & \color{red}{-455x} +819 \color{blue}{+455x} \color{blue}{+84} \\\Leftrightarrow & 494x& = & 903 \\\Leftrightarrow & \frac{494x}{ \color{red}{494} }& = & \frac{903}{494} \\\Leftrightarrow & x = \frac{903}{494} & & \\ & V = \left\{ \frac{903}{494} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & -98x+350 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+98x} & = & \color{red}{-98x} +350 \color{blue}{+98x} \color{blue}{+40} \\\Leftrightarrow & 133x& = & 390 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{390}{133} \\\Leftrightarrow & x = \frac{390}{133} & & \\ & V = \left\{ \frac{390}{133} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & -130x+546 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{+130x} & = & \color{red}{-130x} +546 \color{blue}{+130x} \color{blue}{+18} \\\Leftrightarrow & 169x& = & 564 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{564}{169} \\\Leftrightarrow & x = \frac{564}{169} & & \\ & V = \left\{ \frac{564}{169} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{-7}{4}x+8 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-231}{ \color{blue}{132} }x+\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & -231x+1056 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{+231x} & = & \color{red}{-231x} +1056 \color{blue}{+231x} \color{blue}{+48} \\\Leftrightarrow & 275x& = & 1104 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{1104}{275} \\\Leftrightarrow & x = \frac{1104}{275} & & \\ & V = \left\{ \frac{1104}{275} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{4}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 56x-42 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-56x} & = & \color{red}{56x} -42 \color{blue}{-56x} \color{blue}{+18} \\\Leftrightarrow & -35x& = & -24 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-24}{-35} \\\Leftrightarrow & x = \frac{24}{35} & & \\ & V = \left\{ \frac{24}{35} \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & -88x-396 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{+88x} & = & \color{red}{-88x} -396 \color{blue}{+88x} \color{blue}{-48} \\\Leftrightarrow & 121x& = & -444 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-444}{121} \\\Leftrightarrow & x = \frac{-444}{121} & & \\ & V = \left\{ \frac{-444}{121} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & -224x-672 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{+224x} & = & \color{red}{-224x} -672 \color{blue}{+224x} \color{blue}{-36} \\\Leftrightarrow & 245x& = & -708 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-708}{245} \\\Leftrightarrow & x = \frac{-708}{245} & & \\ & V = \left\{ \frac{-708}{245} \right\} & \\\end{align}\)
8. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-20& = & 65x-130 \\\Leftrightarrow & 26x \color{red}{-20} \color{blue}{+20} \color{blue}{-65x} & = & \color{red}{65x} -130 \color{blue}{-65x} \color{blue}{+20} \\\Leftrightarrow & -39x& = & -110 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-110}{-39} \\\Leftrightarrow & x = \frac{110}{39} & & \\ & V = \left\{ \frac{110}{39} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{-8}{3}x-7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x-\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & -128x-336 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{+128x} & = & \color{red}{-128x} -336 \color{blue}{+128x} \color{blue}{+9} \\\Leftrightarrow & 152x& = & -327 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{-327}{152} \\\Leftrightarrow & x = \frac{-327}{152} & & \\ & V = \left\{ \frac{-327}{152} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & -294x+630 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{+294x} & = & \color{red}{-294x} +630 \color{blue}{+294x} \color{blue}{+45} \\\Leftrightarrow & 329x& = & 675 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{675}{329} \\\Leftrightarrow & x = \frac{675}{329} & & \\ & V = \left\{ \frac{675}{329} \right\} & \\\end{align}\)
11. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-96}{ \color{blue}{120} }x+\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & -96x+600 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{+96x} & = & \color{red}{-96x} +600 \color{blue}{+96x} \color{blue}{-75} \\\Leftrightarrow & 116x& = & 525 \\\Leftrightarrow & \frac{116x}{ \color{red}{116} }& = & \frac{525}{116} \\\Leftrightarrow & x = \frac{525}{116} & & \\ & V = \left\{ \frac{525}{116} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{8}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{36}{ \color{blue}{24} }x-\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+15& = & 36x-48 \\\Leftrightarrow & 8x \color{red}{+15} \color{blue}{-15} \color{blue}{-36x} & = & \color{red}{36x} -48 \color{blue}{-36x} \color{blue}{-15} \\\Leftrightarrow & -28x& = & -63 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{-63}{-28} \\\Leftrightarrow & x = \frac{9}{4} & & \\ & V = \left\{ \frac{9}{4} \right\} & \\\end{align}\)