Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{5}+\frac{5}{8}=\frac{7}{6}x-3\)
2. \(\frac{x}{3}+\frac{4}{7}=\frac{1}{5}x-8\)
3. \(\frac{x}{2}+\frac{2}{15}=\frac{-2}{3}x-2\)
4. \(\frac{x}{6}+\frac{5}{8}=\frac{-2}{5}x-3\)
5. \(\frac{x}{7}+\frac{5}{12}=\frac{3}{2}x+6\)
6. \(\frac{x}{7}+\frac{5}{6}=\frac{5}{4}x-5\)
7. \(\frac{x}{2}+\frac{2}{7}=\frac{-5}{3}x+8\)
8. \(\frac{x}{3}+\frac{4}{13}=\frac{1}{2}x+5\)
9. \(\frac{x}{7}-\frac{5}{16}=\frac{3}{2}x+8\)
10. \(\frac{x}{3}+\frac{4}{15}=\frac{7}{2}x-3\)
11. \(\frac{x}{4}-\frac{2}{15}=\frac{1}{3}x-1\)
12. \(\frac{x}{2}-\frac{3}{13}=\frac{1}{3}x+8\)

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

Verbetersleutel

1. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{7}{6}x-3 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{140}{ \color{blue}{120} }x-\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x+75& = & 140x-360 \\\Leftrightarrow & 24x \color{red}{+75} \color{blue}{-75} \color{blue}{-140x} & = & \color{red}{140x} -360 \color{blue}{-140x} \color{blue}{-75} \\\Leftrightarrow & -116x& = & -435 \\\Leftrightarrow & \frac{-116x}{ \color{red}{-116} }& = & \frac{-435}{-116} \\\Leftrightarrow & x = \frac{15}{4} & & \\ & V = \left\{ \frac{15}{4} \right\} & \\\end{align}\)
2. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{21}{ \color{blue}{105} }x-\frac{840}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+60& = & 21x-840 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-21x} & = & \color{red}{21x} -840 \color{blue}{-21x} \color{blue}{-60} \\\Leftrightarrow & 14x& = & -900 \\\Leftrightarrow & \frac{14x}{ \color{red}{14} }& = & \frac{-900}{14} \\\Leftrightarrow & x = \frac{-450}{7} & & \\ & V = \left\{ \frac{-450}{7} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & -20x-60 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{+20x} & = & \color{red}{-20x} -60 \color{blue}{+20x} \color{blue}{-4} \\\Leftrightarrow & 35x& = & -64 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{-64}{35} \\\Leftrightarrow & x = \frac{-64}{35} & & \\ & V = \left\{ \frac{-64}{35} \right\} & \\\end{align}\)
4. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-48}{ \color{blue}{120} }x-\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & -48x-360 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{+48x} & = & \color{red}{-48x} -360 \color{blue}{+48x} \color{blue}{-75} \\\Leftrightarrow & 68x& = & -435 \\\Leftrightarrow & \frac{68x}{ \color{red}{68} }& = & \frac{-435}{68} \\\Leftrightarrow & x = \frac{-435}{68} & & \\ & V = \left\{ \frac{-435}{68} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{12}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 35 }{ \color{blue}{84} })& = & (\frac{126}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+35& = & 126x+504 \\\Leftrightarrow & 12x \color{red}{+35} \color{blue}{-35} \color{blue}{-126x} & = & \color{red}{126x} +504 \color{blue}{-126x} \color{blue}{-35} \\\Leftrightarrow & -114x& = & 469 \\\Leftrightarrow & \frac{-114x}{ \color{red}{-114} }& = & \frac{469}{-114} \\\Leftrightarrow & x = \frac{-469}{114} & & \\ & V = \left\{ \frac{-469}{114} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 7, 6 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{5}{4}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 70 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+70& = & 105x-420 \\\Leftrightarrow & 12x \color{red}{+70} \color{blue}{-70} \color{blue}{-105x} & = & \color{red}{105x} -420 \color{blue}{-105x} \color{blue}{-70} \\\Leftrightarrow & -93x& = & -490 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{-490}{-93} \\\Leftrightarrow & x = \frac{490}{93} & & \\ & V = \left\{ \frac{490}{93} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -70x+336 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+70x} & = & \color{red}{-70x} +336 \color{blue}{+70x} \color{blue}{-12} \\\Leftrightarrow & 91x& = & 324 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{324}{91} \\\Leftrightarrow & x = \frac{324}{91} & & \\ & V = \left\{ \frac{324}{91} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+24& = & 39x+390 \\\Leftrightarrow & 26x \color{red}{+24} \color{blue}{-24} \color{blue}{-39x} & = & \color{red}{39x} +390 \color{blue}{-39x} \color{blue}{-24} \\\Leftrightarrow & -13x& = & 366 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{366}{-13} \\\Leftrightarrow & x = \frac{-366}{13} & & \\ & V = \left\{ \frac{-366}{13} \right\} & \\\end{align}\)
9. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{3}{2}x+8 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{168}{ \color{blue}{112} }x+\frac{896}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & 168x+896 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{-168x} & = & \color{red}{168x} +896 \color{blue}{-168x} \color{blue}{+35} \\\Leftrightarrow & -152x& = & 931 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{931}{-152} \\\Leftrightarrow & x = \frac{-49}{8} & & \\ & V = \left\{ \frac{-49}{8} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{105}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+8& = & 105x-90 \\\Leftrightarrow & 10x \color{red}{+8} \color{blue}{-8} \color{blue}{-105x} & = & \color{red}{105x} -90 \color{blue}{-105x} \color{blue}{-8} \\\Leftrightarrow & -95x& = & -98 \\\Leftrightarrow & \frac{-95x}{ \color{red}{-95} }& = & \frac{-98}{-95} \\\Leftrightarrow & x = \frac{98}{95} & & \\ & V = \left\{ \frac{98}{95} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{1}{3}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 20x-60 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-20x} & = & \color{red}{20x} -60 \color{blue}{-20x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -52 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-52}{-5} \\\Leftrightarrow & x = \frac{52}{5} & & \\ & V = \left\{ \frac{52}{5} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x+\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & 26x+624 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{-26x} & = & \color{red}{26x} +624 \color{blue}{-26x} \color{blue}{+18} \\\Leftrightarrow & 13x& = & 642 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{642}{13} \\\Leftrightarrow & x = \frac{642}{13} & & \\ & V = \left\{ \frac{642}{13} \right\} & \\\end{align}\)