Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}-\frac{5}{8}=\frac{-7}{4}x+3\)
2. \(\frac{x}{5}+\frac{2}{7}=\frac{5}{6}x+3\)
3. \(\frac{x}{3}+\frac{5}{11}=\frac{-7}{5}x+6\)
4. \(\frac{x}{7}-\frac{3}{7}=\frac{5}{3}x+2\)
5. \(\frac{x}{3}-\frac{2}{15}=\frac{1}{2}x+1\)
6. \(\frac{x}{6}+\frac{4}{15}=\frac{1}{5}x+3\)
7. \(\frac{x}{7}-\frac{2}{11}=\frac{1}{2}x+1\)
8. \(\frac{x}{2}+\frac{3}{16}=\frac{4}{3}x+4\)
9. \(\frac{x}{6}-\frac{5}{6}=\frac{1}{5}x-3\)
10. \(\frac{x}{5}-\frac{5}{7}=\frac{-7}{4}x+1\)
11. \(\frac{x}{3}-\frac{2}{13}=\frac{1}{2}x+5\)
12. \(\frac{x}{6}-\frac{5}{11}=\frac{6}{5}x+2\)

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

Verbetersleutel

1. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{-7}{4}x+3 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-42}{ \color{blue}{24} }x+\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & -42x+72 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{+42x} & = & \color{red}{-42x} +72 \color{blue}{+42x} \color{blue}{+15} \\\Leftrightarrow & 50x& = & 87 \\\Leftrightarrow & \frac{50x}{ \color{red}{50} }& = & \frac{87}{50} \\\Leftrightarrow & x = \frac{87}{50} & & \\ & V = \left\{ \frac{87}{50} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{5}{6}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 175x+630 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-175x} & = & \color{red}{175x} +630 \color{blue}{-175x} \color{blue}{-60} \\\Leftrightarrow & -133x& = & 570 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{570}{-133} \\\Leftrightarrow & x = \frac{-30}{7} & & \\ & V = \left\{ \frac{-30}{7} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{-231}{ \color{blue}{165} }x+\frac{990}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+75& = & -231x+990 \\\Leftrightarrow & 55x \color{red}{+75} \color{blue}{-75} \color{blue}{+231x} & = & \color{red}{-231x} +990 \color{blue}{+231x} \color{blue}{-75} \\\Leftrightarrow & 286x& = & 915 \\\Leftrightarrow & \frac{286x}{ \color{red}{286} }& = & \frac{915}{286} \\\Leftrightarrow & x = \frac{915}{286} & & \\ & V = \left\{ \frac{915}{286} \right\} & \\\end{align}\)
4. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{5}{3}x+2 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 9 }{ \color{blue}{21} })& = & (\frac{35}{ \color{blue}{21} }x+\frac{42}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-9& = & 35x+42 \\\Leftrightarrow & 3x \color{red}{-9} \color{blue}{+9} \color{blue}{-35x} & = & \color{red}{35x} +42 \color{blue}{-35x} \color{blue}{+9} \\\Leftrightarrow & -32x& = & 51 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{51}{-32} \\\Leftrightarrow & x = \frac{-51}{32} & & \\ & V = \left\{ \frac{-51}{32} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 15x+30 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} +30 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 34 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{34}{-5} \\\Leftrightarrow & x = \frac{-34}{5} & & \\ & V = \left\{ \frac{-34}{5} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & 6x+90 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{-6x} & = & \color{red}{6x} +90 \color{blue}{-6x} \color{blue}{-8} \\\Leftrightarrow & -x& = & 82 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{82}{-1} \\\Leftrightarrow & x = -82 & & \\ & V = \left\{ -82 \right\} & \\\end{align}\)
7. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 28 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{154}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-28& = & 77x+154 \\\Leftrightarrow & 22x \color{red}{-28} \color{blue}{+28} \color{blue}{-77x} & = & \color{red}{77x} +154 \color{blue}{-77x} \color{blue}{+28} \\\Leftrightarrow & -55x& = & 182 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{182}{-55} \\\Leftrightarrow & x = \frac{-182}{55} & & \\ & V = \left\{ \frac{-182}{55} \right\} & \\\end{align}\)
8. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{4}{3}x+4 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{64}{ \color{blue}{48} }x+\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 64x+192 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-64x} & = & \color{red}{64x} +192 \color{blue}{-64x} \color{blue}{-9} \\\Leftrightarrow & -40x& = & 183 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{183}{-40} \\\Leftrightarrow & x = \frac{-183}{40} & & \\ & V = \left\{ \frac{-183}{40} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & 6x-90 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{-6x} & = & \color{red}{6x} -90 \color{blue}{-6x} \color{blue}{+25} \\\Leftrightarrow & -x& = & -65 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-65}{-1} \\\Leftrightarrow & x = 65 & & \\ & V = \left\{ 65 \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{7}& = & \frac{-7}{4}x+1 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 100 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x+\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-100& = & -245x+140 \\\Leftrightarrow & 28x \color{red}{-100} \color{blue}{+100} \color{blue}{+245x} & = & \color{red}{-245x} +140 \color{blue}{+245x} \color{blue}{+100} \\\Leftrightarrow & 273x& = & 240 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{240}{273} \\\Leftrightarrow & x = \frac{80}{91} & & \\ & V = \left\{ \frac{80}{91} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-12& = & 39x+390 \\\Leftrightarrow & 26x \color{red}{-12} \color{blue}{+12} \color{blue}{-39x} & = & \color{red}{39x} +390 \color{blue}{-39x} \color{blue}{+12} \\\Leftrightarrow & -13x& = & 402 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{402}{-13} \\\Leftrightarrow & x = \frac{-402}{13} & & \\ & V = \left\{ \frac{-402}{13} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x+\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 396x+660 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-396x} & = & \color{red}{396x} +660 \color{blue}{-396x} \color{blue}{+150} \\\Leftrightarrow & -341x& = & 810 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{810}{-341} \\\Leftrightarrow & x = \frac{-810}{341} & & \\ & V = \left\{ \frac{-810}{341} \right\} & \\\end{align}\)