Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{7}{5}x-8 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }- \frac{ 6 }{ \color{blue}{20} })& = & (\frac{28}{ \color{blue}{20} }x-\frac{160}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x-6& = & 28x-160 \\\Leftrightarrow & 5x \color{red}{-6} \color{blue}{+6} \color{blue}{-28x} & = & \color{red}{28x} -160 \color{blue}{-28x} \color{blue}{+6} \\\Leftrightarrow & -23x& = & -154 \\\Leftrightarrow & \frac{-23x}{ \color{red}{-23} }& = & \frac{-154}{-23} \\\Leftrightarrow & x = \frac{154}{23} & & \\ & V = \left\{ \frac{154}{23} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-640}{ \color{blue}{240} }x-\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-45& = & -640x-1440 \\\Leftrightarrow & 48x \color{red}{-45} \color{blue}{+45} \color{blue}{+640x} & = & \color{red}{-640x} -1440 \color{blue}{+640x} \color{blue}{+45} \\\Leftrightarrow & 688x& = & -1395 \\\Leftrightarrow & \frac{688x}{ \color{red}{688} }& = & \frac{-1395}{688} \\\Leftrightarrow & x = \frac{-1395}{688} & & \\ & V = \left\{ \frac{-1395}{688} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x+\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 40x+240 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-40x} & = & \color{red}{40x} +240 \color{blue}{-40x} \color{blue}{+25} \\\Leftrightarrow & -24x& = & 265 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{265}{-24} \\\Leftrightarrow & x = \frac{-265}{24} & & \\ & V = \left\{ \frac{-265}{24} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & -80x+336 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{+80x} & = & \color{red}{-80x} +336 \color{blue}{+80x} \color{blue}{+15} \\\Leftrightarrow & 104x& = & 351 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{351}{104} \\\Leftrightarrow & x = \frac{27}{8} & & \\ & V = \left\{ \frac{27}{8} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{5}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 140x+420 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-140x} & = & \color{red}{140x} +420 \color{blue}{-140x} \color{blue}{+24} \\\Leftrightarrow & -119x& = & 444 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{444}{-119} \\\Leftrightarrow & x = \frac{-444}{119} & & \\ & V = \left\{ \frac{-444}{119} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -168x-1470 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+168x} & = & \color{red}{-168x} -1470 \color{blue}{+168x} \color{blue}{-60} \\\Leftrightarrow & 203x& = & -1530 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1530}{203} \\\Leftrightarrow & x = \frac{-1530}{203} & & \\ & V = \left\{ \frac{-1530}{203} \right\} & \\\end{align}\)
7. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{4}{5}x+6 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{28}{ \color{blue}{35} }x+\frac{210}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 28x+210 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-28x} & = & \color{red}{28x} +210 \color{blue}{-28x} \color{blue}{-20} \\\Leftrightarrow & -23x& = & 190 \\\Leftrightarrow & \frac{-23x}{ \color{red}{-23} }& = & \frac{190}{-23} \\\Leftrightarrow & x = \frac{-190}{23} & & \\ & V = \left\{ \frac{-190}{23} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{-7}{4}x+2 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-231}{ \color{blue}{132} }x+\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+24& = & -231x+264 \\\Leftrightarrow & 44x \color{red}{+24} \color{blue}{-24} \color{blue}{+231x} & = & \color{red}{-231x} +264 \color{blue}{+231x} \color{blue}{-24} \\\Leftrightarrow & 275x& = & 240 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{240}{275} \\\Leftrightarrow & x = \frac{48}{55} & & \\ & V = \left\{ \frac{48}{55} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & -126x-630 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{+126x} & = & \color{red}{-126x} -630 \color{blue}{+126x} \color{blue}{+50} \\\Leftrightarrow & 141x& = & -580 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{-580}{141} \\\Leftrightarrow & x = \frac{-580}{141} & & \\ & V = \left\{ \frac{-580}{141} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & -56x-420 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{+56x} & = & \color{red}{-56x} -420 \color{blue}{+56x} \color{blue}{+36} \\\Leftrightarrow & 77x& = & -384 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-384}{77} \\\Leftrightarrow & x = \frac{-384}{77} & & \\ & V = \left\{ \frac{-384}{77} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & 12x+180 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-12x} & = & \color{red}{12x} +180 \color{blue}{-12x} \color{blue}{-25} \\\Leftrightarrow & -2x& = & 155 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{155}{-2} \\\Leftrightarrow & x = \frac{-155}{2} & & \\ & V = \left\{ \frac{-155}{2} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-42 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -42 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -66 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-66}{-7} \\\Leftrightarrow & x = \frac{66}{7} & & \\ & V = \left\{ \frac{66}{7} \right\} & \\\end{align}\)