Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 63 }{ \color{blue}{336} })& = & (\frac{280}{ \color{blue}{336} }x+\frac{672}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-63& = & 280x+672 \\\Leftrightarrow & 48x \color{red}{-63} \color{blue}{+63} \color{blue}{-280x} & = & \color{red}{280x} +672 \color{blue}{-280x} \color{blue}{+63} \\\Leftrightarrow & -232x& = & 735 \\\Leftrightarrow & \frac{-232x}{ \color{red}{-232} }& = & \frac{735}{-232} \\\Leftrightarrow & x = \frac{-735}{232} & & \\ & V = \left\{ \frac{-735}{232} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+9& = & -20x+240 \\\Leftrightarrow & 15x \color{red}{+9} \color{blue}{-9} \color{blue}{+20x} & = & \color{red}{-20x} +240 \color{blue}{+20x} \color{blue}{-9} \\\Leftrightarrow & 35x& = & 231 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{231}{35} \\\Leftrightarrow & x = \frac{33}{5} & & \\ & V = \left\{ \frac{33}{5} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -168x+210 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+168x} & = & \color{red}{-168x} +210 \color{blue}{+168x} \color{blue}{-90} \\\Leftrightarrow & 203x& = & 120 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{120}{203} \\\Leftrightarrow & x = \frac{120}{203} & & \\ & V = \left\{ \frac{120}{203} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-4& = & 9x-36 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-9x} & = & \color{red}{9x} -36 \color{blue}{-9x} \color{blue}{+4} \\\Leftrightarrow & -3x& = & -32 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-32}{-3} \\\Leftrightarrow & x = \frac{32}{3} & & \\ & V = \left\{ \frac{32}{3} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 35 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+35& = & 21x-252 \\\Leftrightarrow & 6x \color{red}{+35} \color{blue}{-35} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{-35} \\\Leftrightarrow & -15x& = & -287 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-287}{-15} \\\Leftrightarrow & x = \frac{287}{15} & & \\ & V = \left\{ \frac{287}{15} \right\} & \\\end{align}\)
6. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{8}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x-\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x-9& = & -16x-192 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{+16x} & = & \color{red}{-16x} -192 \color{blue}{+16x} \color{blue}{+9} \\\Leftrightarrow & 22x& = & -183 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{-183}{22} \\\Leftrightarrow & x = \frac{-183}{22} & & \\ & V = \left\{ \frac{-183}{22} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 35x-280 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-35x} & = & \color{red}{35x} -280 \color{blue}{-35x} \color{blue}{+40} \\\Leftrightarrow & -21x& = & -240 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-240}{-21} \\\Leftrightarrow & x = \frac{80}{7} & & \\ & V = \left\{ \frac{80}{7} \right\} & \\\end{align}\)
8. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{105}{ \color{blue}{126} }x+\frac{252}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 105x+252 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-105x} & = & \color{red}{105x} +252 \color{blue}{-105x} \color{blue}{-56} \\\Leftrightarrow & -87x& = & 196 \\\Leftrightarrow & \frac{-87x}{ \color{red}{-87} }& = & \frac{196}{-87} \\\Leftrightarrow & x = \frac{-196}{87} & & \\ & V = \left\{ \frac{-196}{87} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & 196x-84 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{-196x} & = & \color{red}{196x} -84 \color{blue}{-196x} \color{blue}{+36} \\\Leftrightarrow & -175x& = & -48 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-48}{-175} \\\Leftrightarrow & x = \frac{48}{175} & & \\ & V = \left\{ \frac{48}{175} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-12& = & 33x-528 \\\Leftrightarrow & 22x \color{red}{-12} \color{blue}{+12} \color{blue}{-33x} & = & \color{red}{33x} -528 \color{blue}{-33x} \color{blue}{+12} \\\Leftrightarrow & -11x& = & -516 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-516}{-11} \\\Leftrightarrow & x = \frac{516}{11} & & \\ & V = \left\{ \frac{516}{11} \right\} & \\\end{align}\)
11. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x+\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & 40x+640 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-40x} & = & \color{red}{40x} +640 \color{blue}{-40x} \color{blue}{+15} \\\Leftrightarrow & -24x& = & 655 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{655}{-24} \\\Leftrightarrow & x = \frac{-655}{24} & & \\ & V = \left\{ \frac{-655}{24} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 45 }{ \color{blue}{105} })& = & (\frac{126}{ \color{blue}{105} }x-\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+45& = & 126x-525 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{-126x} & = & \color{red}{126x} -525 \color{blue}{-126x} \color{blue}{-45} \\\Leftrightarrow & -91x& = & -570 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-570}{-91} \\\Leftrightarrow & x = \frac{570}{91} & & \\ & V = \left\{ \frac{570}{91} \right\} & \\\end{align}\)