Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{12}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x-25& = & 72x-60 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-72x} & = & \color{red}{72x} -60 \color{blue}{-72x} \color{blue}{+25} \\\Leftrightarrow & -62x& = & -35 \\\Leftrightarrow & \frac{-62x}{ \color{red}{-62} }& = & \frac{-35}{-62} \\\Leftrightarrow & x = \frac{35}{62} & & \\ & V = \left\{ \frac{35}{62} \right\} & \\\end{align}\)
2. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 56 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{462}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-56& = & 77x-462 \\\Leftrightarrow & 22x \color{red}{-56} \color{blue}{+56} \color{blue}{-77x} & = & \color{red}{77x} -462 \color{blue}{-77x} \color{blue}{+56} \\\Leftrightarrow & -55x& = & -406 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-406}{-55} \\\Leftrightarrow & x = \frac{406}{55} & & \\ & V = \left\{ \frac{406}{55} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 7, 6 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 175 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+175& = & 42x-1050 \\\Leftrightarrow & 30x \color{red}{+175} \color{blue}{-175} \color{blue}{-42x} & = & \color{red}{42x} -1050 \color{blue}{-42x} \color{blue}{-175} \\\Leftrightarrow & -12x& = & -1225 \\\Leftrightarrow & \frac{-12x}{ \color{red}{-12} }& = & \frac{-1225}{-12} \\\Leftrightarrow & x = \frac{1225}{12} & & \\ & V = \left\{ \frac{1225}{12} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & 55x+1980 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{-55x} & = & \color{red}{55x} +1980 \color{blue}{-55x} \color{blue}{-150} \\\Leftrightarrow & 11x& = & 1830 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{1830}{11} \\\Leftrightarrow & x = \frac{1830}{11} & & \\ & V = \left\{ \frac{1830}{11} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{5}{6}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+8& = & 25x+60 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-25x} & = & \color{red}{25x} +60 \color{blue}{-25x} \color{blue}{-8} \\\Leftrightarrow & -19x& = & 52 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{52}{-19} \\\Leftrightarrow & x = \frac{-52}{19} & & \\ & V = \left\{ \frac{-52}{19} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 78x+1950 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-78x} & = & \color{red}{78x} +1950 \color{blue}{-78x} \color{blue}{+120} \\\Leftrightarrow & -13x& = & 2070 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{2070}{-13} \\\Leftrightarrow & x = \frac{-2070}{13} & & \\ & V = \left\{ \frac{-2070}{13} \right\} & \\\end{align}\)
7. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{42}{ \color{blue}{35} }x+\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 42x+70 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-42x} & = & \color{red}{42x} +70 \color{blue}{-42x} \color{blue}{-20} \\\Leftrightarrow & -37x& = & 50 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{50}{-37} \\\Leftrightarrow & x = \frac{-50}{37} & & \\ & V = \left\{ \frac{-50}{37} \right\} & \\\end{align}\)
8. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }+ \frac{ 56 }{ \color{blue}{308} })& = & (\frac{-231}{ \color{blue}{308} }x-\frac{2156}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x+56& = & -231x-2156 \\\Leftrightarrow & 44x \color{red}{+56} \color{blue}{-56} \color{blue}{+231x} & = & \color{red}{-231x} -2156 \color{blue}{+231x} \color{blue}{-56} \\\Leftrightarrow & 275x& = & -2212 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{-2212}{275} \\\Leftrightarrow & x = \frac{-2212}{275} & & \\ & V = \left\{ \frac{-2212}{275} \right\} & \\\end{align}\)
9. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x-\frac{12}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x-5& = & 3x-12 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-3x} & = & \color{red}{3x} -12 \color{blue}{-3x} \color{blue}{+5} \\\Leftrightarrow & -x& = & -7 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-7}{-1} \\\Leftrightarrow & x = 7 & & \\ & V = \left\{ 7 \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{3}{4}x+2 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{105}{ \color{blue}{140} }x+\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+60& = & 105x+280 \\\Leftrightarrow & 28x \color{red}{+60} \color{blue}{-60} \color{blue}{-105x} & = & \color{red}{105x} +280 \color{blue}{-105x} \color{blue}{-60} \\\Leftrightarrow & -77x& = & 220 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{220}{-77} \\\Leftrightarrow & x = \frac{-20}{7} & & \\ & V = \left\{ \frac{-20}{7} \right\} & \\\end{align}\)
11. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{7}{4}x-5 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{42}{ \color{blue}{24} }x-\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & 42x-120 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-42x} & = & \color{red}{42x} -120 \color{blue}{-42x} \color{blue}{+15} \\\Leftrightarrow & -34x& = & -105 \\\Leftrightarrow & \frac{-34x}{ \color{red}{-34} }& = & \frac{-105}{-34} \\\Leftrightarrow & x = \frac{105}{34} & & \\ & V = \left\{ \frac{105}{34} \right\} & \\\end{align}\)
12. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{24}{ \color{blue}{120} }x+\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 24x+360 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-24x} & = & \color{red}{24x} +360 \color{blue}{-24x} \color{blue}{+75} \\\Leftrightarrow & -4x& = & 435 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{435}{-4} \\\Leftrightarrow & x = \frac{-435}{4} & & \\ & V = \left\{ \frac{-435}{4} \right\} & \\\end{align}\)