Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & 56x+336 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-56x} & = & \color{red}{56x} +336 \color{blue}{-56x} \color{blue}{-24} \\\Leftrightarrow & -35x& = & 312 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{312}{-35} \\\Leftrightarrow & x = \frac{-312}{35} & & \\ & V = \left\{ \frac{-312}{35} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -168x-1470 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+168x} & = & \color{red}{-168x} -1470 \color{blue}{+168x} \color{blue}{+90} \\\Leftrightarrow & 203x& = & -1380 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1380}{203} \\\Leftrightarrow & x = \frac{-1380}{203} & & \\ & V = \left\{ \frac{-1380}{203} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{6}x+2 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x+\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & 40x+480 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{-40x} & = & \color{red}{40x} +480 \color{blue}{-40x} \color{blue}{+45} \\\Leftrightarrow & 8x& = & 525 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{525}{8} \\\Leftrightarrow & x = \frac{525}{8} & & \\ & V = \left\{ \frac{525}{8} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & -80x-96 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{+80x} & = & \color{red}{-80x} -96 \color{blue}{+80x} \color{blue}{-9} \\\Leftrightarrow & 92x& = & -105 \\\Leftrightarrow & \frac{92x}{ \color{red}{92} }& = & \frac{-105}{92} \\\Leftrightarrow & x = \frac{-105}{92} & & \\ & V = \left\{ \frac{-105}{92} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & 66x+990 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{-66x} & = & \color{red}{66x} +990 \color{blue}{-66x} \color{blue}{-150} \\\Leftrightarrow & -11x& = & 840 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{840}{-11} \\\Leftrightarrow & x = \frac{-840}{11} & & \\ & V = \left\{ \frac{-840}{11} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{5}{4}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+24& = & 105x-252 \\\Leftrightarrow & 28x \color{red}{+24} \color{blue}{-24} \color{blue}{-105x} & = & \color{red}{105x} -252 \color{blue}{-105x} \color{blue}{-24} \\\Leftrightarrow & -77x& = & -276 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-276}{-77} \\\Leftrightarrow & x = \frac{276}{77} & & \\ & V = \left\{ \frac{276}{77} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-42 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -42 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -66 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-66}{-7} \\\Leftrightarrow & x = \frac{66}{7} & & \\ & V = \left\{ \frac{66}{7} \right\} & \\\end{align}\)
8. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 25 }{ \color{blue}{35} })& = & (\frac{-49}{ \color{blue}{35} }x+\frac{35}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+25& = & -49x+35 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+49x} & = & \color{red}{-49x} +35 \color{blue}{+49x} \color{blue}{-25} \\\Leftrightarrow & 54x& = & 10 \\\Leftrightarrow & \frac{54x}{ \color{red}{54} }& = & \frac{10}{54} \\\Leftrightarrow & x = \frac{5}{27} & & \\ & V = \left\{ \frac{5}{27} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{-5}{6}x-4 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x-\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & -75x-360 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{+75x} & = & \color{red}{-75x} -360 \color{blue}{+75x} \color{blue}{+40} \\\Leftrightarrow & 93x& = & -320 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{-320}{93} \\\Leftrightarrow & x = \frac{-320}{93} & & \\ & V = \left\{ \frac{-320}{93} \right\} & \\\end{align}\)
10. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{1}{4}x+7 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{3}{ \color{blue}{12} }x+\frac{84}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & 3x+84 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{-3x} & = & \color{red}{3x} +84 \color{blue}{-3x} \color{blue}{-5} \\\Leftrightarrow & x& = & 79 \\ & V = \left\{ 79 \right\} & \\\end{align}\)
11. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-18}{ \color{blue}{24} }x+\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & -18x+168 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{+18x} & = & \color{red}{-18x} +168 \color{blue}{+18x} \color{blue}{+15} \\\Leftrightarrow & 26x& = & 183 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{183}{26} \\\Leftrightarrow & x = \frac{183}{26} & & \\ & V = \left\{ \frac{183}{26} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 21x-168 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{+35} \\\Leftrightarrow & -15x& = & -133 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-133}{-15} \\\Leftrightarrow & x = \frac{133}{15} & & \\ & V = \left\{ \frac{133}{15} \right\} & \\\end{align}\)