Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 28x+168 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-28x} & = & \color{red}{28x} +168 \color{blue}{-28x} \color{blue}{-36} \\\Leftrightarrow & -7x& = & 132 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{132}{-7} \\\Leftrightarrow & x = \frac{-132}{7} & & \\ & V = \left\{ \frac{-132}{7} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 21x-336 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} -336 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & -7x& = & -318 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-318}{-7} \\\Leftrightarrow & x = \frac{318}{7} & & \\ & V = \left\{ \frac{318}{7} \right\} & \\\end{align}\)
3. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{3}{4}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{99}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-60& = & 99x+660 \\\Leftrightarrow & 44x \color{red}{-60} \color{blue}{+60} \color{blue}{-99x} & = & \color{red}{99x} +660 \color{blue}{-99x} \color{blue}{+60} \\\Leftrightarrow & -55x& = & 720 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{720}{-55} \\\Leftrightarrow & x = \frac{-144}{11} & & \\ & V = \left\{ \frac{-144}{11} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{42}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & 42x-336 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{-42x} & = & \color{red}{42x} -336 \color{blue}{-42x} \color{blue}{+35} \\\Leftrightarrow & -30x& = & -301 \\\Leftrightarrow & \frac{-30x}{ \color{red}{-30} }& = & \frac{-301}{-30} \\\Leftrightarrow & x = \frac{301}{30} & & \\ & V = \left\{ \frac{301}{30} \right\} & \\\end{align}\)
5. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{42}{ \color{blue}{35} }x+\frac{175}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 42x+175 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-42x} & = & \color{red}{42x} +175 \color{blue}{-42x} \color{blue}{-20} \\\Leftrightarrow & -37x& = & 155 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{155}{-37} \\\Leftrightarrow & x = \frac{-155}{37} & & \\ & V = \left\{ \frac{-155}{37} \right\} & \\\end{align}\)
6. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{2}{3}x+1 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{70}{ \color{blue}{105} }x+\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+30& = & 70x+105 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-70x} & = & \color{red}{70x} +105 \color{blue}{-70x} \color{blue}{-30} \\\Leftrightarrow & -49x& = & 75 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{75}{-49} \\\Leftrightarrow & x = \frac{-75}{49} & & \\ & V = \left\{ \frac{-75}{49} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{3}{5}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{36}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 36x-120 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-36x} & = & \color{red}{36x} -120 \color{blue}{-36x} \color{blue}{+8} \\\Leftrightarrow & -21x& = & -112 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-112}{-21} \\\Leftrightarrow & x = \frac{16}{3} & & \\ & V = \left\{ \frac{16}{3} \right\} & \\\end{align}\)
8. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }- \frac{ 30 }{ \color{blue}{195} })& = & (\frac{-273}{ \color{blue}{195} }x-\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x-30& = & -273x-585 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{+273x} & = & \color{red}{-273x} -585 \color{blue}{+273x} \color{blue}{+30} \\\Leftrightarrow & 338x& = & -555 \\\Leftrightarrow & \frac{338x}{ \color{red}{338} }& = & \frac{-555}{338} \\\Leftrightarrow & x = \frac{-555}{338} & & \\ & V = \left\{ \frac{-555}{338} \right\} & \\\end{align}\)
9. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }+ \frac{ 56 }{ \color{blue}{364} })& = & (\frac{-273}{ \color{blue}{364} }x+\frac{1092}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x+56& = & -273x+1092 \\\Leftrightarrow & 52x \color{red}{+56} \color{blue}{-56} \color{blue}{+273x} & = & \color{red}{-273x} +1092 \color{blue}{+273x} \color{blue}{-56} \\\Leftrightarrow & 325x& = & 1036 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{1036}{325} \\\Leftrightarrow & x = \frac{1036}{325} & & \\ & V = \left\{ \frac{1036}{325} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & -312x+1170 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+312x} & = & \color{red}{-312x} +1170 \color{blue}{+312x} \color{blue}{-60} \\\Leftrightarrow & 377x& = & 1110 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{1110}{377} \\\Leftrightarrow & x = \frac{1110}{377} & & \\ & V = \left\{ \frac{1110}{377} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 28x-588 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-28x} & = & \color{red}{28x} -588 \color{blue}{-28x} \color{blue}{+48} \\\Leftrightarrow & -7x& = & -540 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-540}{-7} \\\Leftrightarrow & x = \frac{540}{7} & & \\ & V = \left\{ \frac{540}{7} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 28x+420 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-28x} & = & \color{red}{28x} +420 \color{blue}{-28x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 396 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{396}{-7} \\\Leftrightarrow & x = \frac{-396}{7} & & \\ & V = \left\{ \frac{-396}{7} \right\} & \\\end{align}\)