Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 80 }{ \color{blue}{180} })& = & (\frac{36}{ \color{blue}{180} }x+\frac{1260}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-80& = & 36x+1260 \\\Leftrightarrow & 45x \color{red}{-80} \color{blue}{+80} \color{blue}{-36x} & = & \color{red}{36x} +1260 \color{blue}{-36x} \color{blue}{+80} \\\Leftrightarrow & 9x& = & 1340 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{1340}{9} \\\Leftrightarrow & x = \frac{1340}{9} & & \\ & V = \left\{ \frac{1340}{9} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & 14x+336 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{-14x} & = & \color{red}{14x} +336 \color{blue}{-14x} \color{blue}{+9} \\\Leftrightarrow & 7x& = & 345 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{345}{7} \\\Leftrightarrow & x = \frac{345}{7} & & \\ & V = \left\{ \frac{345}{7} \right\} & \\\end{align}\)
3. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{-7}{4}x+8 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-21}{ \color{blue}{12} }x+\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & -21x+96 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{+21x} & = & \color{red}{-21x} +96 \color{blue}{+21x} \color{blue}{-5} \\\Leftrightarrow & 25x& = & 91 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{91}{25} \\\Leftrightarrow & x = \frac{91}{25} & & \\ & V = \left\{ \frac{91}{25} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{4}{5}x-4 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 100 }{ \color{blue}{220} })& = & (\frac{176}{ \color{blue}{220} }x-\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-100& = & 176x-880 \\\Leftrightarrow & 55x \color{red}{-100} \color{blue}{+100} \color{blue}{-176x} & = & \color{red}{176x} -880 \color{blue}{-176x} \color{blue}{+100} \\\Leftrightarrow & -121x& = & -780 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-780}{-121} \\\Leftrightarrow & x = \frac{780}{121} & & \\ & V = \left\{ \frac{780}{121} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 140x-480 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-140x} & = & \color{red}{140x} -480 \color{blue}{-140x} \color{blue}{+16} \\\Leftrightarrow & -125x& = & -464 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{-464}{-125} \\\Leftrightarrow & x = \frac{464}{125} & & \\ & V = \left\{ \frac{464}{125} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 78x-3120 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-78x} & = & \color{red}{78x} -3120 \color{blue}{-78x} \color{blue}{-90} \\\Leftrightarrow & -13x& = & -3210 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-3210}{-13} \\\Leftrightarrow & x = \frac{3210}{13} & & \\ & V = \left\{ \frac{3210}{13} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{12}& = & \frac{2}{5}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{24}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x-25& = & 24x-180 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-24x} & = & \color{red}{24x} -180 \color{blue}{-24x} \color{blue}{+25} \\\Leftrightarrow & -14x& = & -155 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{-155}{-14} \\\Leftrightarrow & x = \frac{155}{14} & & \\ & V = \left\{ \frac{155}{14} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{4}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 168x-630 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-168x} & = & \color{red}{168x} -630 \color{blue}{-168x} \color{blue}{-90} \\\Leftrightarrow & -133x& = & -720 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-720}{-133} \\\Leftrightarrow & x = \frac{720}{133} & & \\ & V = \left\{ \frac{720}{133} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{5}{6}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-24& = & 35x-126 \\\Leftrightarrow & 6x \color{red}{-24} \color{blue}{+24} \color{blue}{-35x} & = & \color{red}{35x} -126 \color{blue}{-35x} \color{blue}{+24} \\\Leftrightarrow & -29x& = & -102 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-102}{-29} \\\Leftrightarrow & x = \frac{102}{29} & & \\ & V = \left\{ \frac{102}{29} \right\} & \\\end{align}\)
10. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x+\frac{1300}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x+40& = & 65x+1300 \\\Leftrightarrow & 52x \color{red}{+40} \color{blue}{-40} \color{blue}{-65x} & = & \color{red}{65x} +1300 \color{blue}{-65x} \color{blue}{-40} \\\Leftrightarrow & -13x& = & 1260 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{1260}{-13} \\\Leftrightarrow & x = \frac{-1260}{13} & & \\ & V = \left\{ \frac{-1260}{13} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{7}{2}x-2 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{63}{ \color{blue}{18} }x-\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-10& = & 63x-36 \\\Leftrightarrow & 6x \color{red}{-10} \color{blue}{+10} \color{blue}{-63x} & = & \color{red}{63x} -36 \color{blue}{-63x} \color{blue}{+10} \\\Leftrightarrow & -57x& = & -26 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{-26}{-57} \\\Leftrightarrow & x = \frac{26}{57} & & \\ & V = \left\{ \frac{26}{57} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{2}{5}x+1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{36}{ \color{blue}{90} }x+\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & 36x+90 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{-36x} & = & \color{red}{36x} +90 \color{blue}{-36x} \color{blue}{+50} \\\Leftrightarrow & -21x& = & 140 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{140}{-21} \\\Leftrightarrow & x = \frac{-20}{3} & & \\ & V = \left\{ \frac{-20}{3} \right\} & \\\end{align}\)