Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & -196x+700 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{+196x} & = & \color{red}{-196x} +700 \color{blue}{+196x} \color{blue}{+80} \\\Leftrightarrow & 231x& = & 780 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{780}{231} \\\Leftrightarrow & x = \frac{260}{77} & & \\ & V = \left\{ \frac{260}{77} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-36}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -36x-630 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+36x} & = & \color{red}{-36x} -630 \color{blue}{+36x} \color{blue}{+40} \\\Leftrightarrow & 51x& = & -590 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-590}{51} \\\Leftrightarrow & x = \frac{-590}{51} & & \\ & V = \left\{ \frac{-590}{51} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{21}{ \color{blue}{105} }x-\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-45& = & 21x-105 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{-21x} & = & \color{red}{21x} -105 \color{blue}{-21x} \color{blue}{+45} \\\Leftrightarrow & 14x& = & -60 \\\Leftrightarrow & \frac{14x}{ \color{red}{14} }& = & \frac{-60}{14} \\\Leftrightarrow & x = \frac{-30}{7} & & \\ & V = \left\{ \frac{-30}{7} \right\} & \\\end{align}\)
4. \(\text{12 is het kleinste gemene veelvoud van 3, 6 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{7}{4}x+2 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{21}{ \color{blue}{12} }x+\frac{24}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-10& = & 21x+24 \\\Leftrightarrow & 4x \color{red}{-10} \color{blue}{+10} \color{blue}{-21x} & = & \color{red}{21x} +24 \color{blue}{-21x} \color{blue}{+10} \\\Leftrightarrow & -17x& = & 34 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{34}{-17} \\\Leftrightarrow & x = -2 & & \\ & V = \left\{ -2 \right\} & \\\end{align}\)
5. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{8}& = & \frac{5}{3}x-6 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{40}{ \color{blue}{24} }x-\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+9& = & 40x-144 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{-40x} & = & \color{red}{40x} -144 \color{blue}{-40x} \color{blue}{-9} \\\Leftrightarrow & -34x& = & -153 \\\Leftrightarrow & \frac{-34x}{ \color{red}{-34} }& = & \frac{-153}{-34} \\\Leftrightarrow & x = \frac{9}{2} & & \\ & V = \left\{ \frac{9}{2} \right\} & \\\end{align}\)
6. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 50 }{ \color{blue}{130} })& = & (\frac{156}{ \color{blue}{130} }x+\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-50& = & 156x+390 \\\Leftrightarrow & 65x \color{red}{-50} \color{blue}{+50} \color{blue}{-156x} & = & \color{red}{156x} +390 \color{blue}{-156x} \color{blue}{+50} \\\Leftrightarrow & -91x& = & 440 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{440}{-91} \\\Leftrightarrow & x = \frac{-440}{91} & & \\ & V = \left\{ \frac{-440}{91} \right\} & \\\end{align}\)
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. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-18& = & 33x+66 \\\Leftrightarrow & 22x \color{red}{-18} \color{blue}{+18} \color{blue}{-33x} & = & \color{red}{33x} +66 \color{blue}{-33x} \color{blue}{+18} \\\Leftrightarrow & -11x& = & 84 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{84}{-11} \\\Leftrightarrow & x = \frac{-84}{11} & & \\ & V = \left\{ \frac{-84}{11} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{7}{5}x-7 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{168}{ \color{blue}{120} }x-\frac{840}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & 168x-840 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{-168x} & = & \color{red}{168x} -840 \color{blue}{-168x} \color{blue}{-75} \\\Leftrightarrow & -148x& = & -915 \\\Leftrightarrow & \frac{-148x}{ \color{red}{-148} }& = & \frac{-915}{-148} \\\Leftrightarrow & x = \frac{915}{148} & & \\ & V = \left\{ \frac{915}{148} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 35x+350 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-35x} & = & \color{red}{35x} +350 \color{blue}{-35x} \color{blue}{+40} \\\Leftrightarrow & -21x& = & 390 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{390}{-21} \\\Leftrightarrow & x = \frac{-130}{7} & & \\ & V = \left\{ \frac{-130}{7} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 15x+240 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} +240 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -9x& = & 244 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{244}{-9} \\\Leftrightarrow & x = \frac{-244}{9} & & \\ & V = \left\{ \frac{-244}{9} \right\} & \\\end{align}\)
12. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{7}{2}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{231}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 231x+330 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-231x} & = & \color{red}{231x} +330 \color{blue}{-231x} \color{blue}{-30} \\\Leftrightarrow & -209x& = & 300 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{300}{-209} \\\Leftrightarrow & x = \frac{-300}{209} & & \\ & V = \left\{ \frac{-300}{209} \right\} & \\\end{align}\)