Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{5}{3}x-1 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{80}{ \color{blue}{48} }x-\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 80x-48 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-80x} & = & \color{red}{80x} -48 \color{blue}{-80x} \color{blue}{-15} \\\Leftrightarrow & -56x& = & -63 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{-63}{-56} \\\Leftrightarrow & x = \frac{9}{8} & & \\ & V = \left\{ \frac{9}{8} \right\} & \\\end{align}\)
2. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{12}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-32}{ \color{blue}{12} }x+\frac{24}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x-5& = & -32x+24 \\\Leftrightarrow & 6x \color{red}{-5} \color{blue}{+5} \color{blue}{+32x} & = & \color{red}{-32x} +24 \color{blue}{+32x} \color{blue}{+5} \\\Leftrightarrow & 38x& = & 29 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{29}{38} \\\Leftrightarrow & x = \frac{29}{38} & & \\ & V = \left\{ \frac{29}{38} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & -72x+720 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{+72x} & = & \color{red}{-72x} +720 \color{blue}{+72x} \color{blue}{+20} \\\Leftrightarrow & 117x& = & 740 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{740}{117} \\\Leftrightarrow & x = \frac{740}{117} & & \\ & V = \left\{ \frac{740}{117} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & -40x+60 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{+40x} & = & \color{red}{-40x} +60 \color{blue}{+40x} \color{blue}{-18} \\\Leftrightarrow & 55x& = & 42 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{42}{55} \\\Leftrightarrow & x = \frac{42}{55} & & \\ & V = \left\{ \frac{42}{55} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x+1470 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} +1470 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & 1530 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{1530}{203} \\\Leftrightarrow & x = \frac{1530}{203} & & \\ & V = \left\{ \frac{1530}{203} \right\} & \\\end{align}\)
6. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & -130x-312 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{+130x} & = & \color{red}{-130x} -312 \color{blue}{+130x} \color{blue}{-12} \\\Leftrightarrow & 169x& = & -324 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-324}{169} \\\Leftrightarrow & x = \frac{-324}{169} & & \\ & V = \left\{ \frac{-324}{169} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & -12x-240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{+12x} & = & \color{red}{-12x} -240 \color{blue}{+12x} \color{blue}{+8} \\\Leftrightarrow & 27x& = & -232 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{-232}{27} \\\Leftrightarrow & x = \frac{-232}{27} & & \\ & V = \left\{ \frac{-232}{27} \right\} & \\\end{align}\)
8. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-325}{ \color{blue}{195} }x-\frac{780}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+45& = & -325x-780 \\\Leftrightarrow & 39x \color{red}{+45} \color{blue}{-45} \color{blue}{+325x} & = & \color{red}{-325x} -780 \color{blue}{+325x} \color{blue}{-45} \\\Leftrightarrow & 364x& = & -825 \\\Leftrightarrow & \frac{364x}{ \color{red}{364} }& = & \frac{-825}{364} \\\Leftrightarrow & x = \frac{-825}{364} & & \\ & V = \left\{ \frac{-825}{364} \right\} & \\\end{align}\)
9. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }- \frac{ 60 }{ \color{blue}{195} })& = & (\frac{-156}{ \color{blue}{195} }x+\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x-60& = & -156x+585 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{+156x} & = & \color{red}{-156x} +585 \color{blue}{+156x} \color{blue}{+60} \\\Leftrightarrow & 221x& = & 645 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{645}{221} \\\Leftrightarrow & x = \frac{645}{221} & & \\ & V = \left\{ \frac{645}{221} \right\} & \\\end{align}\)
10. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x+\frac{840}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-45& = & 35x+840 \\\Leftrightarrow & 21x \color{red}{-45} \color{blue}{+45} \color{blue}{-35x} & = & \color{red}{35x} +840 \color{blue}{-35x} \color{blue}{+45} \\\Leftrightarrow & -14x& = & 885 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{885}{-14} \\\Leftrightarrow & x = \frac{-885}{14} & & \\ & V = \left\{ \frac{-885}{14} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{3}{2}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{117}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+18& = & 117x+156 \\\Leftrightarrow & 26x \color{red}{+18} \color{blue}{-18} \color{blue}{-117x} & = & \color{red}{117x} +156 \color{blue}{-117x} \color{blue}{-18} \\\Leftrightarrow & -91x& = & 138 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{138}{-91} \\\Leftrightarrow & x = \frac{-138}{91} & & \\ & V = \left\{ \frac{-138}{91} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{45}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 45x-180 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-45x} & = & \color{red}{45x} -180 \color{blue}{-45x} \color{blue}{+9} \\\Leftrightarrow & -35x& = & -171 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-171}{-35} \\\Leftrightarrow & x = \frac{171}{35} & & \\ & V = \left\{ \frac{171}{35} \right\} & \\\end{align}\)