Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{12}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{4}{ \color{blue}{12} }x+\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x+5& = & 4x+48 \\\Leftrightarrow & 6x \color{red}{+5} \color{blue}{-5} \color{blue}{-4x} & = & \color{red}{4x} +48 \color{blue}{-4x} \color{blue}{-5} \\\Leftrightarrow & 2x& = & 43 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{43}{2} \\\Leftrightarrow & x = \frac{43}{2} & & \\ & V = \left\{ \frac{43}{2} \right\} & \\\end{align}\)
2. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{4}{5}x-8 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{88}{ \color{blue}{110} }x-\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-40& = & 88x-880 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-88x} & = & \color{red}{88x} -880 \color{blue}{-88x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & -840 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-840}{-33} \\\Leftrightarrow & x = \frac{280}{11} & & \\ & V = \left\{ \frac{280}{11} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & -196x-280 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{+196x} & = & \color{red}{-196x} -280 \color{blue}{+196x} \color{blue}{+80} \\\Leftrightarrow & 231x& = & -200 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-200}{231} \\\Leftrightarrow & x = \frac{-200}{231} & & \\ & V = \left\{ \frac{-200}{231} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & -24x-90 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{+24x} & = & \color{red}{-24x} -90 \color{blue}{+24x} \color{blue}{+9} \\\Leftrightarrow & 34x& = & -81 \\\Leftrightarrow & \frac{34x}{ \color{red}{34} }& = & \frac{-81}{34} \\\Leftrightarrow & x = \frac{-81}{34} & & \\ & V = \left\{ \frac{-81}{34} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{4}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{12}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 12x+384 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} +384 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & 4x& = & 393 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{393}{4} \\\Leftrightarrow & x = \frac{393}{4} & & \\ & V = \left\{ \frac{393}{4} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{4}{3}x+3 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{16}{ \color{blue}{12} }x+\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+5& = & 16x+36 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-16x} & = & \color{red}{16x} +36 \color{blue}{-16x} \color{blue}{-5} \\\Leftrightarrow & -13x& = & 31 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{31}{-13} \\\Leftrightarrow & x = \frac{-31}{13} & & \\ & V = \left\{ \frac{-31}{13} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-208}{ \color{blue}{78} }x+\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-12& = & -208x+546 \\\Leftrightarrow & 39x \color{red}{-12} \color{blue}{+12} \color{blue}{+208x} & = & \color{red}{-208x} +546 \color{blue}{+208x} \color{blue}{+12} \\\Leftrightarrow & 247x& = & 558 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{558}{247} \\\Leftrightarrow & x = \frac{558}{247} & & \\ & V = \left\{ \frac{558}{247} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+40& = & 35x-70 \\\Leftrightarrow & 14x \color{red}{+40} \color{blue}{-40} \color{blue}{-35x} & = & \color{red}{35x} -70 \color{blue}{-35x} \color{blue}{-40} \\\Leftrightarrow & -21x& = & -110 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-110}{-21} \\\Leftrightarrow & x = \frac{110}{21} & & \\ & V = \left\{ \frac{110}{21} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & 312x+780 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{-312x} & = & \color{red}{312x} +780 \color{blue}{-312x} \color{blue}{-120} \\\Leftrightarrow & -247x& = & 660 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{660}{-247} \\\Leftrightarrow & x = \frac{-660}{247} & & \\ & V = \left\{ \frac{-660}{247} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x+\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & -176x+396 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{+176x} & = & \color{red}{-176x} +396 \color{blue}{+176x} \color{blue}{-12} \\\Leftrightarrow & 209x& = & 384 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{384}{209} \\\Leftrightarrow & x = \frac{384}{209} & & \\ & V = \left\{ \frac{384}{209} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x+\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & -84x+300 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+84x} & = & \color{red}{-84x} +300 \color{blue}{+84x} \color{blue}{+8} \\\Leftrightarrow & 99x& = & 308 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{308}{99} \\\Leftrightarrow & x = \frac{28}{9} & & \\ & V = \left\{ \frac{28}{9} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{40}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 40x+240 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-40x} & = & \color{red}{40x} +240 \color{blue}{-40x} \color{blue}{-8} \\\Leftrightarrow & -25x& = & 232 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{232}{-25} \\\Leftrightarrow & x = \frac{-232}{25} & & \\ & V = \left\{ \frac{-232}{25} \right\} & \\\end{align}\)