Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & -42x+120 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{+42x} & = & \color{red}{-42x} +120 \color{blue}{+42x} \color{blue}{+4} \\\Leftrightarrow & 57x& = & 124 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{124}{57} \\\Leftrightarrow & x = \frac{124}{57} & & \\ & V = \left\{ \frac{124}{57} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & 70x+120 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-70x} & = & \color{red}{70x} +120 \color{blue}{-70x} \color{blue}{+9} \\\Leftrightarrow & -64x& = & 129 \\\Leftrightarrow & \frac{-64x}{ \color{red}{-64} }& = & \frac{129}{-64} \\\Leftrightarrow & x = \frac{-129}{64} & & \\ & V = \left\{ \frac{-129}{64} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x-\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-60& = & 35x-735 \\\Leftrightarrow & 21x \color{red}{-60} \color{blue}{+60} \color{blue}{-35x} & = & \color{red}{35x} -735 \color{blue}{-35x} \color{blue}{+60} \\\Leftrightarrow & -14x& = & -675 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{-675}{-14} \\\Leftrightarrow & x = \frac{675}{14} & & \\ & V = \left\{ \frac{675}{14} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 78x+3120 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-78x} & = & \color{red}{78x} +3120 \color{blue}{-78x} \color{blue}{+120} \\\Leftrightarrow & -13x& = & 3240 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{3240}{-13} \\\Leftrightarrow & x = \frac{-3240}{13} & & \\ & V = \left\{ \frac{-3240}{13} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x-1260 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} -1260 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & -1200 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1200}{203} \\\Leftrightarrow & x = \frac{-1200}{203} & & \\ & V = \left\{ \frac{-1200}{203} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-96}{ \color{blue}{36} }x+\frac{252}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & -96x+252 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+96x} & = & \color{red}{-96x} +252 \color{blue}{+96x} \color{blue}{-8} \\\Leftrightarrow & 105x& = & 244 \\\Leftrightarrow & \frac{105x}{ \color{red}{105} }& = & \frac{244}{105} \\\Leftrightarrow & x = \frac{244}{105} & & \\ & V = \left\{ \frac{244}{105} \right\} & \\\end{align}\)
7. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x-\frac{280}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & -28x-280 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{+28x} & = & \color{red}{-28x} -280 \color{blue}{+28x} \color{blue}{-20} \\\Leftrightarrow & 33x& = & -300 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-300}{33} \\\Leftrightarrow & x = \frac{-100}{11} & & \\ & V = \left\{ \frac{-100}{11} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x+\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-24& = & 44x+1056 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{-44x} & = & \color{red}{44x} +1056 \color{blue}{-44x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & 1080 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{1080}{-11} \\\Leftrightarrow & x = \frac{-1080}{11} & & \\ & V = \left\{ \frac{-1080}{11} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x+42 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} +42 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & 30 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{30}{-7} \\\Leftrightarrow & x = \frac{-30}{7} & & \\ & V = \left\{ \frac{-30}{7} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{2}{3}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{32}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+15& = & 32x+384 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{-32x} & = & \color{red}{32x} +384 \color{blue}{-32x} \color{blue}{-15} \\\Leftrightarrow & -20x& = & 369 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{369}{-20} \\\Leftrightarrow & x = \frac{-369}{20} & & \\ & V = \left\{ \frac{-369}{20} \right\} & \\\end{align}\)
11. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{55}{ \color{blue}{165} }x+\frac{825}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-75& = & 55x+825 \\\Leftrightarrow & 33x \color{red}{-75} \color{blue}{+75} \color{blue}{-55x} & = & \color{red}{55x} +825 \color{blue}{-55x} \color{blue}{+75} \\\Leftrightarrow & -22x& = & 900 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{900}{-22} \\\Leftrightarrow & x = \frac{-450}{11} & & \\ & V = \left\{ \frac{-450}{11} \right\} & \\\end{align}\)
12. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x-\frac{42}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x-5& = & 3x-42 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-3x} & = & \color{red}{3x} -42 \color{blue}{-3x} \color{blue}{+5} \\\Leftrightarrow & -x& = & -37 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-37}{-1} \\\Leftrightarrow & x = 37 & & \\ & V = \left\{ 37 \right\} & \\\end{align}\)