Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-8& = & 15x-180 \\\Leftrightarrow & 10x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} -180 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -172 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-172}{-5} \\\Leftrightarrow & x = \frac{172}{5} & & \\ & V = \left\{ \frac{172}{5} \right\} & \\\end{align}\)
2. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{-264}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & -264x+1980 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{+264x} & = & \color{red}{-264x} +1980 \color{blue}{+264x} \color{blue}{-90} \\\Leftrightarrow & 319x& = & 1890 \\\Leftrightarrow & \frac{319x}{ \color{red}{319} }& = & \frac{1890}{319} \\\Leftrightarrow & x = \frac{1890}{319} & & \\ & V = \left\{ \frac{1890}{319} \right\} & \\\end{align}\)
3. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{7}{6}x-8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 28 }{ \color{blue}{126} })& = & (\frac{147}{ \color{blue}{126} }x-\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-28& = & 147x-1008 \\\Leftrightarrow & 18x \color{red}{-28} \color{blue}{+28} \color{blue}{-147x} & = & \color{red}{147x} -1008 \color{blue}{-147x} \color{blue}{+28} \\\Leftrightarrow & -129x& = & -980 \\\Leftrightarrow & \frac{-129x}{ \color{red}{-129} }& = & \frac{-980}{-129} \\\Leftrightarrow & x = \frac{980}{129} & & \\ & V = \left\{ \frac{980}{129} \right\} & \\\end{align}\)
4. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 10 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x-\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+10& = & -28x-70 \\\Leftrightarrow & 5x \color{red}{+10} \color{blue}{-10} \color{blue}{+28x} & = & \color{red}{-28x} -70 \color{blue}{+28x} \color{blue}{-10} \\\Leftrightarrow & 33x& = & -80 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-80}{33} \\\Leftrightarrow & x = \frac{-80}{33} & & \\ & V = \left\{ \frac{-80}{33} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{1}{3}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 16x-96 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-16x} & = & \color{red}{16x} -96 \color{blue}{-16x} \color{blue}{-15} \\\Leftrightarrow & 8x& = & -111 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-111}{8} \\\Leftrightarrow & x = \frac{-111}{8} & & \\ & V = \left\{ \frac{-111}{8} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{4}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 168x-1260 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-168x} & = & \color{red}{168x} -1260 \color{blue}{-168x} \color{blue}{+120} \\\Leftrightarrow & -133x& = & -1140 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-1140}{-133} \\\Leftrightarrow & x = \frac{60}{7} & & \\ & V = \left\{ \frac{60}{7} \right\} & \\\end{align}\)
7. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 12x-168 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} -168 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & -4x& = & -159 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-159}{-4} \\\Leftrightarrow & x = \frac{159}{4} & & \\ & V = \left\{ \frac{159}{4} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -56x-420 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+56x} & = & \color{red}{-56x} -420 \color{blue}{+56x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & -396 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-396}{77} \\\Leftrightarrow & x = \frac{-36}{7} & & \\ & V = \left\{ \frac{-36}{7} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x+252 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} +252 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 276 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{276}{-7} \\\Leftrightarrow & x = \frac{-276}{7} & & \\ & V = \left\{ \frac{-276}{7} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & 24x+192 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-24x} & = & \color{red}{24x} +192 \color{blue}{-24x} \color{blue}{+15} \\\Leftrightarrow & -8x& = & 207 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{207}{-8} \\\Leftrightarrow & x = \frac{-207}{8} & & \\ & V = \left\{ \frac{-207}{8} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 56 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-56& = & 35x-1260 \\\Leftrightarrow & 30x \color{red}{-56} \color{blue}{+56} \color{blue}{-35x} & = & \color{red}{35x} -1260 \color{blue}{-35x} \color{blue}{+56} \\\Leftrightarrow & -5x& = & -1204 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-1204}{-5} \\\Leftrightarrow & x = \frac{1204}{5} & & \\ & V = \left\{ \frac{1204}{5} \right\} & \\\end{align}\)
12. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{14}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 5 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{56}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-5& = & 7x+56 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-7x} & = & \color{red}{7x} +56 \color{blue}{-7x} \color{blue}{+5} \\\Leftrightarrow & -5x& = & 61 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{61}{-5} \\\Leftrightarrow & x = \frac{-61}{5} & & \\ & V = \left\{ \frac{-61}{5} \right\} & \\\end{align}\)