Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}+\frac{4}{7}=\frac{7}{2}x-3\)
2. \(\frac{x}{7}+\frac{5}{7}=\frac{-7}{4}x+1\)
3. \(\frac{x}{4}+\frac{2}{11}=\frac{-7}{5}x+4\)
4. \(\frac{x}{2}+\frac{4}{15}=\frac{1}{3}x-3\)
5. \(\frac{x}{5}+\frac{4}{7}=\frac{-5}{6}x+2\)
6. \(\frac{x}{5}+\frac{3}{7}=\frac{1}{2}x-1\)
7. \(\frac{x}{6}+\frac{3}{14}=\frac{4}{5}x-1\)
8. \(\frac{x}{7}+\frac{3}{7}=\frac{-4}{5}x-3\)
9. \(\frac{x}{6}+\frac{3}{13}=\frac{2}{5}x-3\)
10. \(\frac{x}{2}-\frac{3}{10}=\frac{7}{3}x+5\)
11. \(\frac{x}{6}-\frac{2}{7}=\frac{-4}{5}x+5\)
12. \(\frac{x}{6}+\frac{3}{8}=\frac{2}{5}x+4\)

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 147x-126 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-147x} & = & \color{red}{147x} -126 \color{blue}{-147x} \color{blue}{-24} \\\Leftrightarrow & -133x& = & -150 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-150}{-133} \\\Leftrightarrow & x = \frac{150}{133} & & \\ & V = \left\{ \frac{150}{133} \right\} & \\\end{align}\)
2. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{-7}{4}x+1 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 20 }{ \color{blue}{28} })& = & (\frac{-49}{ \color{blue}{28} }x+\frac{28}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+20& = & -49x+28 \\\Leftrightarrow & 4x \color{red}{+20} \color{blue}{-20} \color{blue}{+49x} & = & \color{red}{-49x} +28 \color{blue}{+49x} \color{blue}{-20} \\\Leftrightarrow & 53x& = & 8 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{8}{53} \\\Leftrightarrow & x = \frac{8}{53} & & \\ & V = \left\{ \frac{8}{53} \right\} & \\\end{align}\)
3. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 40 }{ \color{blue}{220} })& = & (\frac{-308}{ \color{blue}{220} }x+\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+40& = & -308x+880 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{+308x} & = & \color{red}{-308x} +880 \color{blue}{+308x} \color{blue}{-40} \\\Leftrightarrow & 363x& = & 840 \\\Leftrightarrow & \frac{363x}{ \color{red}{363} }& = & \frac{840}{363} \\\Leftrightarrow & x = \frac{280}{121} & & \\ & V = \left\{ \frac{280}{121} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 10x-90 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-10x} & = & \color{red}{10x} -90 \color{blue}{-10x} \color{blue}{-8} \\\Leftrightarrow & 5x& = & -98 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{-98}{5} \\\Leftrightarrow & x = \frac{-98}{5} & & \\ & V = \left\{ \frac{-98}{5} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{-5}{6}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & -175x+420 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{+175x} & = & \color{red}{-175x} +420 \color{blue}{+175x} \color{blue}{-120} \\\Leftrightarrow & 217x& = & 300 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{300}{217} \\\Leftrightarrow & x = \frac{300}{217} & & \\ & V = \left\{ \frac{300}{217} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+30& = & 35x-70 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-35x} & = & \color{red}{35x} -70 \color{blue}{-35x} \color{blue}{-30} \\\Leftrightarrow & -21x& = & -100 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-100}{-21} \\\Leftrightarrow & x = \frac{100}{21} & & \\ & V = \left\{ \frac{100}{21} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{4}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & 168x-210 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{-168x} & = & \color{red}{168x} -210 \color{blue}{-168x} \color{blue}{-45} \\\Leftrightarrow & -133x& = & -255 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-255}{-133} \\\Leftrightarrow & x = \frac{255}{133} & & \\ & V = \left\{ \frac{255}{133} \right\} & \\\end{align}\)
8. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 15 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x-\frac{105}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+15& = & -28x-105 \\\Leftrightarrow & 5x \color{red}{+15} \color{blue}{-15} \color{blue}{+28x} & = & \color{red}{-28x} -105 \color{blue}{+28x} \color{blue}{-15} \\\Leftrightarrow & 33x& = & -120 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-120}{33} \\\Leftrightarrow & x = \frac{-40}{11} & & \\ & V = \left\{ \frac{-40}{11} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{2}{5}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{156}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 156x-1170 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-156x} & = & \color{red}{156x} -1170 \color{blue}{-156x} \color{blue}{-90} \\\Leftrightarrow & -91x& = & -1260 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-1260}{-91} \\\Leftrightarrow & x = \frac{180}{13} & & \\ & V = \left\{ \frac{180}{13} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & 70x+150 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{-70x} & = & \color{red}{70x} +150 \color{blue}{-70x} \color{blue}{+9} \\\Leftrightarrow & -55x& = & 159 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{159}{-55} \\\Leftrightarrow & x = \frac{-159}{55} & & \\ & V = \left\{ \frac{-159}{55} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x+1050 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} +1050 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & 1110 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{1110}{203} \\\Leftrightarrow & x = \frac{1110}{203} & & \\ & V = \left\{ \frac{1110}{203} \right\} & \\\end{align}\)
12. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{8}& = & \frac{2}{5}x+4 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{48}{ \color{blue}{120} }x+\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+45& = & 48x+480 \\\Leftrightarrow & 20x \color{red}{+45} \color{blue}{-45} \color{blue}{-48x} & = & \color{red}{48x} +480 \color{blue}{-48x} \color{blue}{-45} \\\Leftrightarrow & -28x& = & 435 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{435}{-28} \\\Leftrightarrow & x = \frac{-435}{28} & & \\ & V = \left\{ \frac{-435}{28} \right\} & \\\end{align}\)