Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -70x-42 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+70x} & = & \color{red}{-70x} -42 \color{blue}{+70x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & -24 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-24}{91} \\\Leftrightarrow & x = \frac{-24}{91} & & \\ & V = \left\{ \frac{-24}{91} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{7}{2}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{105}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 105x+90 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-105x} & = & \color{red}{105x} +90 \color{blue}{-105x} \color{blue}{+4} \\\Leftrightarrow & -95x& = & 94 \\\Leftrightarrow & \frac{-95x}{ \color{red}{-95} }& = & \frac{94}{-95} \\\Leftrightarrow & x = \frac{-94}{95} & & \\ & V = \left\{ \frac{-94}{95} \right\} & \\\end{align}\)
3. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x-144 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} -144 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & -153 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-153}{-8} \\\Leftrightarrow & x = \frac{153}{8} & & \\ & V = \left\{ \frac{153}{8} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{7}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{546}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 546x+3120 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-546x} & = & \color{red}{546x} +3120 \color{blue}{-546x} \color{blue}{+120} \\\Leftrightarrow & -481x& = & 3240 \\\Leftrightarrow & \frac{-481x}{ \color{red}{-481} }& = & \frac{3240}{-481} \\\Leftrightarrow & x = \frac{-3240}{481} & & \\ & V = \left\{ \frac{-3240}{481} \right\} & \\\end{align}\)
5. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{7}{2}x-2 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 21 }{ \color{blue}{112} })& = & (\frac{392}{ \color{blue}{112} }x-\frac{224}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-21& = & 392x-224 \\\Leftrightarrow & 16x \color{red}{-21} \color{blue}{+21} \color{blue}{-392x} & = & \color{red}{392x} -224 \color{blue}{-392x} \color{blue}{+21} \\\Leftrightarrow & -376x& = & -203 \\\Leftrightarrow & \frac{-376x}{ \color{red}{-376} }& = & \frac{-203}{-376} \\\Leftrightarrow & x = \frac{203}{376} & & \\ & V = \left\{ \frac{203}{376} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 33x-66 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-33x} & = & \color{red}{33x} -66 \color{blue}{-33x} \color{blue}{-30} \\\Leftrightarrow & -11x& = & -96 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-96}{-11} \\\Leftrightarrow & x = \frac{96}{11} & & \\ & V = \left\{ \frac{96}{11} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x-\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & 14x-70 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{-14x} & = & \color{red}{14x} -70 \color{blue}{-14x} \color{blue}{+21} \\\Leftrightarrow & -4x& = & -49 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-49}{-4} \\\Leftrightarrow & x = \frac{49}{4} & & \\ & V = \left\{ \frac{49}{4} \right\} & \\\end{align}\)
8. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-30}{ \color{blue}{18} }x+\frac{144}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & -30x+144 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{+30x} & = & \color{red}{-30x} +144 \color{blue}{+30x} \color{blue}{+4} \\\Leftrightarrow & 39x& = & 148 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{148}{39} \\\Leftrightarrow & x = \frac{148}{39} & & \\ & V = \left\{ \frac{148}{39} \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-280}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+60& = & -280x+210 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{+280x} & = & \color{red}{-280x} +210 \color{blue}{+280x} \color{blue}{-60} \\\Leftrightarrow & 301x& = & 150 \\\Leftrightarrow & \frac{301x}{ \color{red}{301} }& = & \frac{150}{301} \\\Leftrightarrow & x = \frac{150}{301} & & \\ & V = \left\{ \frac{150}{301} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+15& = & 24x+288 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-24x} & = & \color{red}{24x} +288 \color{blue}{-24x} \color{blue}{-15} \\\Leftrightarrow & -8x& = & 273 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{273}{-8} \\\Leftrightarrow & x = \frac{-273}{8} & & \\ & V = \left\{ \frac{-273}{8} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-12& = & -130x+624 \\\Leftrightarrow & 39x \color{red}{-12} \color{blue}{+12} \color{blue}{+130x} & = & \color{red}{-130x} +624 \color{blue}{+130x} \color{blue}{+12} \\\Leftrightarrow & 169x& = & 636 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{636}{169} \\\Leftrightarrow & x = \frac{636}{169} & & \\ & V = \left\{ \frac{636}{169} \right\} & \\\end{align}\)
12. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{7}{4}x-4 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{21}{ \color{blue}{12} }x-\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & 21x-48 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{-21x} & = & \color{red}{21x} -48 \color{blue}{-21x} \color{blue}{-5} \\\Leftrightarrow & -17x& = & -53 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{-53}{-17} \\\Leftrightarrow & x = \frac{53}{17} & & \\ & V = \left\{ \frac{53}{17} \right\} & \\\end{align}\)