Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & -24x-90 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{+24x} & = & \color{red}{-24x} -90 \color{blue}{+24x} \color{blue}{+4} \\\Leftrightarrow & 29x& = & -86 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-86}{29} \\\Leftrightarrow & x = \frac{-86}{29} & & \\ & V = \left\{ \frac{-86}{29} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-24& = & 39x-312 \\\Leftrightarrow & 26x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} -312 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & -288 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-288}{-13} \\\Leftrightarrow & x = \frac{288}{13} & & \\ & V = \left\{ \frac{288}{13} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -168x+1050 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+168x} & = & \color{red}{-168x} +1050 \color{blue}{+168x} \color{blue}{-120} \\\Leftrightarrow & 203x& = & 930 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{930}{203} \\\Leftrightarrow & x = \frac{930}{203} & & \\ & V = \left\{ \frac{930}{203} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 21x+210 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} +210 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & -7x& = & 228 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{228}{-7} \\\Leftrightarrow & x = \frac{-228}{7} & & \\ & V = \left\{ \frac{-228}{7} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 14x-252 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-14x} & = & \color{red}{14x} -252 \color{blue}{-14x} \color{blue}{+12} \\\Leftrightarrow & 7x& = & -240 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-240}{7} \\\Leftrightarrow & x = \frac{-240}{7} & & \\ & V = \left\{ \frac{-240}{7} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 33x-66 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-33x} & = & \color{red}{33x} -66 \color{blue}{-33x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & -90 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-90}{-11} \\\Leftrightarrow & x = \frac{90}{11} & & \\ & V = \left\{ \frac{90}{11} \right\} & \\\end{align}\)
7. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x-\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 6x-72 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} -72 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & 3x& = & -76 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-76}{3} \\\Leftrightarrow & x = \frac{-76}{3} & & \\ & V = \left\{ \frac{-76}{3} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{5}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 140x+672 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-140x} & = & \color{red}{140x} +672 \color{blue}{-140x} \color{blue}{-48} \\\Leftrightarrow & -119x& = & 624 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{624}{-119} \\\Leftrightarrow & x = \frac{-624}{119} & & \\ & V = \left\{ \frac{-624}{119} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{7}{6}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{385}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & 385x+2640 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{-385x} & = & \color{red}{385x} +2640 \color{blue}{-385x} \color{blue}{-120} \\\Leftrightarrow & -319x& = & 2520 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{2520}{-319} \\\Leftrightarrow & x = \frac{-2520}{319} & & \\ & V = \left\{ \frac{-2520}{319} \right\} & \\\end{align}\)
10. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{3}{4}x-5 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{117}{ \color{blue}{156} }x-\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & 117x-780 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{-117x} & = & \color{red}{117x} -780 \color{blue}{-117x} \color{blue}{+36} \\\Leftrightarrow & -65x& = & -744 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-744}{-65} \\\Leftrightarrow & x = \frac{744}{65} & & \\ & V = \left\{ \frac{744}{65} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 78x+1560 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-78x} & = & \color{red}{78x} +1560 \color{blue}{-78x} \color{blue}{-90} \\\Leftrightarrow & -13x& = & 1470 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{1470}{-13} \\\Leftrightarrow & x = \frac{-1470}{13} & & \\ & V = \left\{ \frac{-1470}{13} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{7}{6}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-120& = & 245x+1680 \\\Leftrightarrow & 42x \color{red}{-120} \color{blue}{+120} \color{blue}{-245x} & = & \color{red}{245x} +1680 \color{blue}{-245x} \color{blue}{+120} \\\Leftrightarrow & -203x& = & 1800 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{1800}{-203} \\\Leftrightarrow & x = \frac{-1800}{203} & & \\ & V = \left\{ \frac{-1800}{203} \right\} & \\\end{align}\)