Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-8}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -224x+672 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+224x} & = & \color{red}{-224x} +672 \color{blue}{+224x} \color{blue}{-48} \\\Leftrightarrow & 245x& = & 624 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{624}{245} \\\Leftrightarrow & x = \frac{624}{245} & & \\ & V = \left\{ \frac{624}{245} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x+\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & 12x+48 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-12x} & = & \color{red}{12x} +48 \color{blue}{-12x} \color{blue}{+15} \\\Leftrightarrow & -4x& = & 63 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{63}{-4} \\\Leftrightarrow & x = \frac{-63}{4} & & \\ & V = \left\{ \frac{-63}{4} \right\} & \\\end{align}\)
3. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{5}{6}x-4 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 84 }{ \color{blue}{462} })& = & (\frac{385}{ \color{blue}{462} }x-\frac{1848}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+84& = & 385x-1848 \\\Leftrightarrow & 66x \color{red}{+84} \color{blue}{-84} \color{blue}{-385x} & = & \color{red}{385x} -1848 \color{blue}{-385x} \color{blue}{-84} \\\Leftrightarrow & -319x& = & -1932 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{-1932}{-319} \\\Leftrightarrow & x = \frac{1932}{319} & & \\ & V = \left\{ \frac{1932}{319} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-352}{ \color{blue}{132} }x-\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & -352x-1056 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{+352x} & = & \color{red}{-352x} -1056 \color{blue}{+352x} \color{blue}{-48} \\\Leftrightarrow & 385x& = & -1104 \\\Leftrightarrow & \frac{385x}{ \color{red}{385} }& = & \frac{-1104}{385} \\\Leftrightarrow & x = \frac{-1104}{385} & & \\ & V = \left\{ \frac{-1104}{385} \right\} & \\\end{align}\)
5. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{8}& = & \frac{-5}{6}x-6 \\\Leftrightarrow & \color{blue}{168.} (\frac{24x}{ \color{blue}{168} }- \frac{ 105 }{ \color{blue}{168} })& = & (\frac{-140}{ \color{blue}{168} }x-\frac{1008}{ \color{blue}{168} }) \color{blue}{.168} \\\Leftrightarrow & 24x-105& = & -140x-1008 \\\Leftrightarrow & 24x \color{red}{-105} \color{blue}{+105} \color{blue}{+140x} & = & \color{red}{-140x} -1008 \color{blue}{+140x} \color{blue}{+105} \\\Leftrightarrow & 164x& = & -903 \\\Leftrightarrow & \frac{164x}{ \color{red}{164} }& = & \frac{-903}{164} \\\Leftrightarrow & x = \frac{-903}{164} & & \\ & V = \left\{ \frac{-903}{164} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{7}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{-147}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+60& = & -147x-252 \\\Leftrightarrow & 28x \color{red}{+60} \color{blue}{-60} \color{blue}{+147x} & = & \color{red}{-147x} -252 \color{blue}{+147x} \color{blue}{-60} \\\Leftrightarrow & 175x& = & -312 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-312}{175} \\\Leftrightarrow & x = \frac{-312}{175} & & \\ & V = \left\{ \frac{-312}{175} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{7}{2}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{105}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+8& = & 105x-30 \\\Leftrightarrow & 10x \color{red}{+8} \color{blue}{-8} \color{blue}{-105x} & = & \color{red}{105x} -30 \color{blue}{-105x} \color{blue}{-8} \\\Leftrightarrow & -95x& = & -38 \\\Leftrightarrow & \frac{-95x}{ \color{red}{-95} }& = & \frac{-38}{-95} \\\Leftrightarrow & x = \frac{2}{5} & & \\ & V = \left\{ \frac{2}{5} \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 7x-112 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-7x} & = & \color{red}{7x} -112 \color{blue}{-7x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -104 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-104}{-5} \\\Leftrightarrow & x = \frac{104}{5} & & \\ & V = \left\{ \frac{104}{5} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-273}{ \color{blue}{156} }x-\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & -273x-468 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{+273x} & = & \color{red}{-273x} -468 \color{blue}{+273x} \color{blue}{+36} \\\Leftrightarrow & 325x& = & -432 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{-432}{325} \\\Leftrightarrow & x = \frac{-432}{325} & & \\ & V = \left\{ \frac{-432}{325} \right\} & \\\end{align}\)
10. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{5}{2}x-1 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 21 }{ \color{blue}{56} })& = & (\frac{140}{ \color{blue}{56} }x-\frac{56}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+21& = & 140x-56 \\\Leftrightarrow & 8x \color{red}{+21} \color{blue}{-21} \color{blue}{-140x} & = & \color{red}{140x} -56 \color{blue}{-140x} \color{blue}{-21} \\\Leftrightarrow & -132x& = & -77 \\\Leftrightarrow & \frac{-132x}{ \color{red}{-132} }& = & \frac{-77}{-132} \\\Leftrightarrow & x = \frac{7}{12} & & \\ & V = \left\{ \frac{7}{12} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{5}{3}x+5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{260}{ \color{blue}{156} }x+\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+24& = & 260x+780 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-260x} & = & \color{red}{260x} +780 \color{blue}{-260x} \color{blue}{-24} \\\Leftrightarrow & -221x& = & 756 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{756}{-221} \\\Leftrightarrow & x = \frac{-756}{221} & & \\ & V = \left\{ \frac{-756}{221} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x-\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & 55x-1320 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{-55x} & = & \color{red}{55x} -1320 \color{blue}{-55x} \color{blue}{-120} \\\Leftrightarrow & 11x& = & -1440 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-1440}{11} \\\Leftrightarrow & x = \frac{-1440}{11} & & \\ & V = \left\{ \frac{-1440}{11} \right\} & \\\end{align}\)