Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 252x-630 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-252x} & = & \color{red}{252x} -630 \color{blue}{-252x} \color{blue}{+120} \\\Leftrightarrow & -217x& = & -510 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-510}{-217} \\\Leftrightarrow & x = \frac{510}{217} & & \\ & V = \left\{ \frac{510}{217} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & -140x-588 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{+140x} & = & \color{red}{-140x} -588 \color{blue}{+140x} \color{blue}{-36} \\\Leftrightarrow & 161x& = & -624 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-624}{161} \\\Leftrightarrow & x = \frac{-624}{161} & & \\ & V = \left\{ \frac{-624}{161} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 7, 6 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 70 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+70& = & -63x-168 \\\Leftrightarrow & 12x \color{red}{+70} \color{blue}{-70} \color{blue}{+63x} & = & \color{red}{-63x} -168 \color{blue}{+63x} \color{blue}{-70} \\\Leftrightarrow & 75x& = & -238 \\\Leftrightarrow & \frac{75x}{ \color{red}{75} }& = & \frac{-238}{75} \\\Leftrightarrow & x = \frac{-238}{75} & & \\ & V = \left\{ \frac{-238}{75} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+9& = & -20x-30 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{+20x} & = & \color{red}{-20x} -30 \color{blue}{+20x} \color{blue}{-9} \\\Leftrightarrow & 26x& = & -39 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-39}{26} \\\Leftrightarrow & x = \frac{-3}{2} & & \\ & V = \left\{ \frac{-3}{2} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -294x+840 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+294x} & = & \color{red}{-294x} +840 \color{blue}{+294x} \color{blue}{+90} \\\Leftrightarrow & 329x& = & 930 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{930}{329} \\\Leftrightarrow & x = \frac{930}{329} & & \\ & V = \left\{ \frac{930}{329} \right\} & \\\end{align}\)
6. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{5}{3}x-7 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 12 }{ \color{blue}{21} })& = & (\frac{35}{ \color{blue}{21} }x-\frac{147}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-12& = & 35x-147 \\\Leftrightarrow & 3x \color{red}{-12} \color{blue}{+12} \color{blue}{-35x} & = & \color{red}{35x} -147 \color{blue}{-35x} \color{blue}{+12} \\\Leftrightarrow & -32x& = & -135 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{-135}{-32} \\\Leftrightarrow & x = \frac{135}{32} & & \\ & V = \left\{ \frac{135}{32} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & 42x+1470 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{-42x} & = & \color{red}{42x} +1470 \color{blue}{-42x} \color{blue}{+45} \\\Leftrightarrow & -7x& = & 1515 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{1515}{-7} \\\Leftrightarrow & x = \frac{-1515}{7} & & \\ & V = \left\{ \frac{-1515}{7} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{2}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{36}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 36x-630 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-36x} & = & \color{red}{36x} -630 \color{blue}{-36x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & -670 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-670}{-21} \\\Leftrightarrow & x = \frac{670}{21} & & \\ & V = \left\{ \frac{670}{21} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -70x-42 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+70x} & = & \color{red}{-70x} -42 \color{blue}{+70x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & -18 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-18}{91} \\\Leftrightarrow & x = \frac{-18}{91} & & \\ & V = \left\{ \frac{-18}{91} \right\} & \\\end{align}\)
10. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{7}{4}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{273}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+48& = & 273x-156 \\\Leftrightarrow & 52x \color{red}{+48} \color{blue}{-48} \color{blue}{-273x} & = & \color{red}{273x} -156 \color{blue}{-273x} \color{blue}{-48} \\\Leftrightarrow & -221x& = & -204 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{-204}{-221} \\\Leftrightarrow & x = \frac{12}{13} & & \\ & V = \left\{ \frac{12}{13} \right\} & \\\end{align}\)
11. \(\text{40 is het kleinste gemene veelvoud van 4, 8 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{8}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{40.} (\frac{10x}{ \color{blue}{40} }- \frac{ 25 }{ \color{blue}{40} })& = & (\frac{8}{ \color{blue}{40} }x+\frac{240}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 10x-25& = & 8x+240 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-8x} & = & \color{red}{8x} +240 \color{blue}{-8x} \color{blue}{+25} \\\Leftrightarrow & 2x& = & 265 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{265}{2} \\\Leftrightarrow & x = \frac{265}{2} & & \\ & V = \left\{ \frac{265}{2} \right\} & \\\end{align}\)
12. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{-10}{ \color{blue}{6} }x+\frac{42}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & -10x+42 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{+10x} & = & \color{red}{-10x} +42 \color{blue}{+10x} \color{blue}{-5} \\\Leftrightarrow & 13x& = & 37 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{37}{13} \\\Leftrightarrow & x = \frac{37}{13} & & \\ & V = \left\{ \frac{37}{13} \right\} & \\\end{align}\)