Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}+\frac{4}{11}=\frac{7}{5}x-7\)
2. \(\frac{x}{5}-\frac{3}{13}=\frac{1}{2}x-6\)
3. \(\frac{x}{4}+\frac{4}{7}=\frac{-8}{3}x+6\)
4. \(\frac{x}{3}+\frac{2}{15}=\frac{1}{5}x-5\)
5. \(\frac{x}{7}-\frac{3}{14}=\frac{1}{2}x+6\)
6. \(\frac{x}{7}+\frac{4}{13}=\frac{-5}{6}x+2\)
7. \(\frac{x}{2}+\frac{3}{16}=\frac{7}{3}x-7\)
8. \(\frac{x}{2}-\frac{5}{12}=\frac{1}{3}x-5\)
9. \(\frac{x}{3}+\frac{3}{7}=\frac{7}{2}x+8\)
10. \(\frac{x}{3}-\frac{5}{8}=\frac{1}{2}x-5\)
11. \(\frac{x}{2}-\frac{2}{7}=\frac{-8}{3}x-6\)
12. \(\frac{x}{4}-\frac{2}{9}=\frac{6}{5}x-6\)

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

Verbetersleutel

1. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{7}{5}x-7 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 60 }{ \color{blue}{165} })& = & (\frac{231}{ \color{blue}{165} }x-\frac{1155}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+60& = & 231x-1155 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{-231x} & = & \color{red}{231x} -1155 \color{blue}{-231x} \color{blue}{-60} \\\Leftrightarrow & -176x& = & -1215 \\\Leftrightarrow & \frac{-176x}{ \color{red}{-176} }& = & \frac{-1215}{-176} \\\Leftrightarrow & x = \frac{1215}{176} & & \\ & V = \left\{ \frac{1215}{176} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-30& = & 65x-780 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-65x} & = & \color{red}{65x} -780 \color{blue}{-65x} \color{blue}{+30} \\\Leftrightarrow & -39x& = & -750 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-750}{-39} \\\Leftrightarrow & x = \frac{250}{13} & & \\ & V = \left\{ \frac{250}{13} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -224x+504 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+224x} & = & \color{red}{-224x} +504 \color{blue}{+224x} \color{blue}{-48} \\\Leftrightarrow & 245x& = & 456 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{456}{245} \\\Leftrightarrow & x = \frac{456}{245} & & \\ & V = \left\{ \frac{456}{245} \right\} & \\\end{align}\)
4. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 2 }{ \color{blue}{15} })& = & (\frac{3}{ \color{blue}{15} }x-\frac{75}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+2& = & 3x-75 \\\Leftrightarrow & 5x \color{red}{+2} \color{blue}{-2} \color{blue}{-3x} & = & \color{red}{3x} -75 \color{blue}{-3x} \color{blue}{-2} \\\Leftrightarrow & 2x& = & -77 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{-77}{2} \\\Leftrightarrow & x = \frac{-77}{2} & & \\ & V = \left\{ \frac{-77}{2} \right\} & \\\end{align}\)
5. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 3 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-3& = & 7x+84 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-7x} & = & \color{red}{7x} +84 \color{blue}{-7x} \color{blue}{+3} \\\Leftrightarrow & -5x& = & 87 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{87}{-5} \\\Leftrightarrow & x = \frac{-87}{5} & & \\ & V = \left\{ \frac{-87}{5} \right\} & \\\end{align}\)
6. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{-5}{6}x+2 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 168 }{ \color{blue}{546} })& = & (\frac{-455}{ \color{blue}{546} }x+\frac{1092}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+168& = & -455x+1092 \\\Leftrightarrow & 78x \color{red}{+168} \color{blue}{-168} \color{blue}{+455x} & = & \color{red}{-455x} +1092 \color{blue}{+455x} \color{blue}{-168} \\\Leftrightarrow & 533x& = & 924 \\\Leftrightarrow & \frac{533x}{ \color{red}{533} }& = & \frac{924}{533} \\\Leftrightarrow & x = \frac{924}{533} & & \\ & V = \left\{ \frac{924}{533} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x-\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 112x-336 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-112x} & = & \color{red}{112x} -336 \color{blue}{-112x} \color{blue}{-9} \\\Leftrightarrow & -88x& = & -345 \\\Leftrightarrow & \frac{-88x}{ \color{red}{-88} }& = & \frac{-345}{-88} \\\Leftrightarrow & x = \frac{345}{88} & & \\ & V = \left\{ \frac{345}{88} \right\} & \\\end{align}\)
8. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{12}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{4}{ \color{blue}{12} }x-\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x-5& = & 4x-60 \\\Leftrightarrow & 6x \color{red}{-5} \color{blue}{+5} \color{blue}{-4x} & = & \color{red}{4x} -60 \color{blue}{-4x} \color{blue}{+5} \\\Leftrightarrow & 2x& = & -55 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{-55}{2} \\\Leftrightarrow & x = \frac{-55}{2} & & \\ & V = \left\{ \frac{-55}{2} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{7}{2}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 147x+336 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-147x} & = & \color{red}{147x} +336 \color{blue}{-147x} \color{blue}{-18} \\\Leftrightarrow & -133x& = & 318 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{318}{-133} \\\Leftrightarrow & x = \frac{-318}{133} & & \\ & V = \left\{ \frac{-318}{133} \right\} & \\\end{align}\)
10. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & 12x-120 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-12x} & = & \color{red}{12x} -120 \color{blue}{-12x} \color{blue}{+15} \\\Leftrightarrow & -4x& = & -105 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-105}{-4} \\\Leftrightarrow & x = \frac{105}{4} & & \\ & V = \left\{ \frac{105}{4} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -112x-252 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+112x} & = & \color{red}{-112x} -252 \color{blue}{+112x} \color{blue}{+12} \\\Leftrightarrow & 133x& = & -240 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-240}{133} \\\Leftrightarrow & x = \frac{-240}{133} & & \\ & V = \left\{ \frac{-240}{133} \right\} & \\\end{align}\)
12. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{6}{5}x-6 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 40 }{ \color{blue}{180} })& = & (\frac{216}{ \color{blue}{180} }x-\frac{1080}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-40& = & 216x-1080 \\\Leftrightarrow & 45x \color{red}{-40} \color{blue}{+40} \color{blue}{-216x} & = & \color{red}{216x} -1080 \color{blue}{-216x} \color{blue}{+40} \\\Leftrightarrow & -171x& = & -1040 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{-1040}{-171} \\\Leftrightarrow & x = \frac{1040}{171} & & \\ & V = \left\{ \frac{1040}{171} \right\} & \\\end{align}\)