Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{5}{2}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & 175x+140 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{-175x} & = & \color{red}{175x} +140 \color{blue}{-175x} \color{blue}{+21} \\\Leftrightarrow & -165x& = & 161 \\\Leftrightarrow & \frac{-165x}{ \color{red}{-165} }& = & \frac{161}{-165} \\\Leftrightarrow & x = \frac{-161}{165} & & \\ & V = \left\{ \frac{-161}{165} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{7}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 294x-1050 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-294x} & = & \color{red}{294x} -1050 \color{blue}{-294x} \color{blue}{+120} \\\Leftrightarrow & -259x& = & -930 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{-930}{-259} \\\Leftrightarrow & x = \frac{930}{259} & & \\ & V = \left\{ \frac{930}{259} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-8}{3}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-80}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & -80x-90 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{+80x} & = & \color{red}{-80x} -90 \color{blue}{+80x} \color{blue}{+9} \\\Leftrightarrow & 95x& = & -81 \\\Leftrightarrow & \frac{95x}{ \color{red}{95} }& = & \frac{-81}{95} \\\Leftrightarrow & x = \frac{-81}{95} & & \\ & V = \left\{ \frac{-81}{95} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & -224x-672 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{+224x} & = & \color{red}{-224x} -672 \color{blue}{+224x} \color{blue}{+36} \\\Leftrightarrow & 245x& = & -636 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-636}{245} \\\Leftrightarrow & x = \frac{-636}{245} & & \\ & V = \left\{ \frac{-636}{245} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{3}{2}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 63x+336 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-63x} & = & \color{red}{63x} +336 \color{blue}{-63x} \color{blue}{-24} \\\Leftrightarrow & -49x& = & 312 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{312}{-49} \\\Leftrightarrow & x = \frac{-312}{49} & & \\ & V = \left\{ \frac{-312}{49} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 15x-150 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} -150 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & -146 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-146}{-5} \\\Leftrightarrow & x = \frac{146}{5} & & \\ & V = \left\{ \frac{146}{5} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -72x+180 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+72x} & = & \color{red}{-72x} +180 \color{blue}{+72x} \color{blue}{-40} \\\Leftrightarrow & 87x& = & 140 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{140}{87} \\\Leftrightarrow & x = \frac{140}{87} & & \\ & V = \left\{ \frac{140}{87} \right\} & \\\end{align}\)
8. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 55x+660 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-55x} & = & \color{red}{55x} +660 \color{blue}{-55x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & 700 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{700}{-33} \\\Leftrightarrow & x = \frac{-700}{33} & & \\ & V = \left\{ \frac{-700}{33} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x+\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & 468x+390 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{-468x} & = & \color{red}{468x} +390 \color{blue}{-468x} \color{blue}{-120} \\\Leftrightarrow & -403x& = & 270 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{270}{-403} \\\Leftrightarrow & x = \frac{-270}{403} & & \\ & V = \left\{ \frac{-270}{403} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{315}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 315x-630 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-315x} & = & \color{red}{315x} -630 \color{blue}{-315x} \color{blue}{+20} \\\Leftrightarrow & -297x& = & -610 \\\Leftrightarrow & \frac{-297x}{ \color{red}{-297} }& = & \frac{-610}{-297} \\\Leftrightarrow & x = \frac{610}{297} & & \\ & V = \left\{ \frac{610}{297} \right\} & \\\end{align}\)
11. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 70 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{770}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-70& = & 77x+770 \\\Leftrightarrow & 22x \color{red}{-70} \color{blue}{+70} \color{blue}{-77x} & = & \color{red}{77x} +770 \color{blue}{-77x} \color{blue}{+70} \\\Leftrightarrow & -55x& = & 840 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{840}{-55} \\\Leftrightarrow & x = \frac{-168}{11} & & \\ & V = \left\{ \frac{-168}{11} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{30}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & 30x-120 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{-30x} & = & \color{red}{30x} -120 \color{blue}{-30x} \color{blue}{-25} \\\Leftrightarrow & -18x& = & -145 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{-145}{-18} \\\Leftrightarrow & x = \frac{145}{18} & & \\ & V = \left\{ \frac{145}{18} \right\} & \\\end{align}\)