Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{1}{4}x-2 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 12 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x-\frac{56}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+12& = & 7x-56 \\\Leftrightarrow & 4x \color{red}{+12} \color{blue}{-12} \color{blue}{-7x} & = & \color{red}{7x} -56 \color{blue}{-7x} \color{blue}{-12} \\\Leftrightarrow & -3x& = & -68 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-68}{-3} \\\Leftrightarrow & x = \frac{68}{3} & & \\ & V = \left\{ \frac{68}{3} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -140x-252 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+140x} & = & \color{red}{-140x} -252 \color{blue}{+140x} \color{blue}{+48} \\\Leftrightarrow & 161x& = & -204 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-204}{161} \\\Leftrightarrow & x = \frac{-204}{161} & & \\ & V = \left\{ \frac{-204}{161} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x+\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & -30x+108 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+30x} & = & \color{red}{-30x} +108 \color{blue}{+30x} \color{blue}{+8} \\\Leftrightarrow & 39x& = & 116 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{116}{39} \\\Leftrightarrow & x = \frac{116}{39} & & \\ & V = \left\{ \frac{116}{39} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{13}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-150& = & 78x+780 \\\Leftrightarrow & 65x \color{red}{-150} \color{blue}{+150} \color{blue}{-78x} & = & \color{red}{78x} +780 \color{blue}{-78x} \color{blue}{+150} \\\Leftrightarrow & -13x& = & 930 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{930}{-13} \\\Leftrightarrow & x = \frac{-930}{13} & & \\ & V = \left\{ \frac{-930}{13} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & 35x-1470 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{-35x} & = & \color{red}{35x} -1470 \color{blue}{-35x} \color{blue}{-90} \\\Leftrightarrow & 7x& = & -1560 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1560}{7} \\\Leftrightarrow & x = \frac{-1560}{7} & & \\ & V = \left\{ \frac{-1560}{7} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 42x+630 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-42x} & = & \color{red}{42x} +630 \color{blue}{-42x} \color{blue}{-60} \\\Leftrightarrow & -7x& = & 570 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{570}{-7} \\\Leftrightarrow & x = \frac{-570}{7} & & \\ & V = \left\{ \frac{-570}{7} \right\} & \\\end{align}\)
7. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{144}{ \color{blue}{120} }x+\frac{120}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & 144x+120 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{-144x} & = & \color{red}{144x} +120 \color{blue}{-144x} \color{blue}{-75} \\\Leftrightarrow & -124x& = & 45 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{45}{-124} \\\Leftrightarrow & x = \frac{-45}{124} & & \\ & V = \left\{ \frac{-45}{124} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{3}{4}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{117}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+24& = & 117x-156 \\\Leftrightarrow & 52x \color{red}{+24} \color{blue}{-24} \color{blue}{-117x} & = & \color{red}{117x} -156 \color{blue}{-117x} \color{blue}{-24} \\\Leftrightarrow & -65x& = & -180 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-180}{-65} \\\Leftrightarrow & x = \frac{36}{13} & & \\ & V = \left\{ \frac{36}{13} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 28x-84 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-28x} & = & \color{red}{28x} -84 \color{blue}{-28x} \color{blue}{-36} \\\Leftrightarrow & -7x& = & -120 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-120}{-7} \\\Leftrightarrow & x = \frac{120}{7} & & \\ & V = \left\{ \frac{120}{7} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & -44x-264 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{+44x} & = & \color{red}{-44x} -264 \color{blue}{+44x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & -240 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-240}{77} \\\Leftrightarrow & x = \frac{-240}{77} & & \\ & V = \left\{ \frac{-240}{77} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x+210 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} +210 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & 190 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{190}{-21} \\\Leftrightarrow & x = \frac{-190}{21} & & \\ & V = \left\{ \frac{-190}{21} \right\} & \\\end{align}\)
12. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{9}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 100 }{ \color{blue}{180} })& = & (\frac{36}{ \color{blue}{180} }x-\frac{360}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+100& = & 36x-360 \\\Leftrightarrow & 45x \color{red}{+100} \color{blue}{-100} \color{blue}{-36x} & = & \color{red}{36x} -360 \color{blue}{-36x} \color{blue}{-100} \\\Leftrightarrow & 9x& = & -460 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{-460}{9} \\\Leftrightarrow & x = \frac{-460}{9} & & \\ & V = \left\{ \frac{-460}{9} \right\} & \\\end{align}\)