Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-16& = & -45x+180 \\\Leftrightarrow & 20x \color{red}{-16} \color{blue}{+16} \color{blue}{+45x} & = & \color{red}{-45x} +180 \color{blue}{+45x} \color{blue}{+16} \\\Leftrightarrow & 65x& = & 196 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{196}{65} \\\Leftrightarrow & x = \frac{196}{65} & & \\ & V = \left\{ \frac{196}{65} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 10 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+10& = & 7x+112 \\\Leftrightarrow & 2x \color{red}{+10} \color{blue}{-10} \color{blue}{-7x} & = & \color{red}{7x} +112 \color{blue}{-7x} \color{blue}{-10} \\\Leftrightarrow & -5x& = & 102 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{102}{-5} \\\Leftrightarrow & x = \frac{-102}{5} & & \\ & V = \left\{ \frac{-102}{5} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x+\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-60& = & 35x+735 \\\Leftrightarrow & 21x \color{red}{-60} \color{blue}{+60} \color{blue}{-35x} & = & \color{red}{35x} +735 \color{blue}{-35x} \color{blue}{+60} \\\Leftrightarrow & -14x& = & 795 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{795}{-14} \\\Leftrightarrow & x = \frac{-795}{14} & & \\ & V = \left\{ \frac{-795}{14} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-15& = & 16x+384 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} +384 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & -4x& = & 399 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{399}{-4} \\\Leftrightarrow & x = \frac{-399}{4} & & \\ & V = \left\{ \frac{-399}{4} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -72x+90 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+72x} & = & \color{red}{-72x} +90 \color{blue}{+72x} \color{blue}{+40} \\\Leftrightarrow & 87x& = & 130 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{130}{87} \\\Leftrightarrow & x = \frac{130}{87} & & \\ & V = \left\{ \frac{130}{87} \right\} & \\\end{align}\)
6. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-60}{ \color{blue}{80} }x+\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & -60x+80 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{+60x} & = & \color{red}{-60x} +80 \color{blue}{+60x} \color{blue}{-25} \\\Leftrightarrow & 76x& = & 55 \\\Leftrightarrow & \frac{76x}{ \color{red}{76} }& = & \frac{55}{76} \\\Leftrightarrow & x = \frac{55}{76} & & \\ & V = \left\{ \frac{55}{76} \right\} & \\\end{align}\)
7. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{-30}{ \color{blue}{45} }x+\frac{315}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-20& = & -30x+315 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{+30x} & = & \color{red}{-30x} +315 \color{blue}{+30x} \color{blue}{+20} \\\Leftrightarrow & 39x& = & 335 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{335}{39} \\\Leftrightarrow & x = \frac{335}{39} & & \\ & V = \left\{ \frac{335}{39} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{3}{2}x+4 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{99}{ \color{blue}{66} }x+\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-18& = & 99x+264 \\\Leftrightarrow & 22x \color{red}{-18} \color{blue}{+18} \color{blue}{-99x} & = & \color{red}{99x} +264 \color{blue}{-99x} \color{blue}{+18} \\\Leftrightarrow & -77x& = & 282 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{282}{-77} \\\Leftrightarrow & x = \frac{-282}{77} & & \\ & V = \left\{ \frac{-282}{77} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{7}{3}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 98x-126 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-98x} & = & \color{red}{98x} -126 \color{blue}{-98x} \color{blue}{+24} \\\Leftrightarrow & -77x& = & -102 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-102}{-77} \\\Leftrightarrow & x = \frac{102}{77} & & \\ & V = \left\{ \frac{102}{77} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{12}& = & \frac{-7}{4}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 35 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+35& = & -147x-84 \\\Leftrightarrow & 12x \color{red}{+35} \color{blue}{-35} \color{blue}{+147x} & = & \color{red}{-147x} -84 \color{blue}{+147x} \color{blue}{-35} \\\Leftrightarrow & 159x& = & -119 \\\Leftrightarrow & \frac{159x}{ \color{red}{159} }& = & \frac{-119}{159} \\\Leftrightarrow & x = \frac{-119}{159} & & \\ & V = \left\{ \frac{-119}{159} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & -175x-1470 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{+175x} & = & \color{red}{-175x} -1470 \color{blue}{+175x} \color{blue}{+60} \\\Leftrightarrow & 217x& = & -1410 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-1410}{217} \\\Leftrightarrow & x = \frac{-1410}{217} & & \\ & V = \left\{ \frac{-1410}{217} \right\} & \\\end{align}\)
12. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x-\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-10& = & -30x-54 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{+30x} & = & \color{red}{-30x} -54 \color{blue}{+30x} \color{blue}{+10} \\\Leftrightarrow & 39x& = & -44 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-44}{39} \\\Leftrightarrow & x = \frac{-44}{39} & & \\ & V = \left\{ \frac{-44}{39} \right\} & \\\end{align}\)