Vgln. eerste graad (reeks 4)

Hoofdmenu Eentje per keer  🖨

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

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

---

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 7, 6 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 175 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-175& = & 252x+1260 \\\Leftrightarrow & 30x \color{red}{-175} \color{blue}{+175} \color{blue}{-252x} & = & \color{red}{252x} +1260 \color{blue}{-252x} \color{blue}{+175} \\\Leftrightarrow & -222x& = & 1435 \\\Leftrightarrow & \frac{-222x}{ \color{red}{-222} }& = & \frac{1435}{-222} \\\Leftrightarrow & x = \frac{-1435}{222} & & \\ & V = \left\{ \frac{-1435}{222} \right\} & \\\end{align}\)
2. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x+\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+40& = & -105x+1120 \\\Leftrightarrow & 28x \color{red}{+40} \color{blue}{-40} \color{blue}{+105x} & = & \color{red}{-105x} +1120 \color{blue}{+105x} \color{blue}{-40} \\\Leftrightarrow & 133x& = & 1080 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{1080}{133} \\\Leftrightarrow & x = \frac{1080}{133} & & \\ & V = \left\{ \frac{1080}{133} \right\} & \\\end{align}\)
3. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{5}{2}x+1 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{315}{ \color{blue}{126} }x+\frac{126}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 315x+126 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-315x} & = & \color{red}{315x} +126 \color{blue}{-315x} \color{blue}{-56} \\\Leftrightarrow & -297x& = & 70 \\\Leftrightarrow & \frac{-297x}{ \color{red}{-297} }& = & \frac{70}{-297} \\\Leftrightarrow & x = \frac{-70}{297} & & \\ & V = \left\{ \frac{-70}{297} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -40x+120 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+40x} & = & \color{red}{-40x} +120 \color{blue}{+40x} \color{blue}{-16} \\\Leftrightarrow & 55x& = & 104 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{104}{55} \\\Leftrightarrow & x = \frac{104}{55} & & \\ & V = \left\{ \frac{104}{55} \right\} & \\\end{align}\)
5. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{-2}{3}x+4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{-12}{ \color{blue}{18} }x+\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & -12x+72 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{+12x} & = & \color{red}{-12x} +72 \color{blue}{+12x} \color{blue}{+4} \\\Leftrightarrow & 21x& = & 76 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{76}{21} \\\Leftrightarrow & x = \frac{76}{21} & & \\ & V = \left\{ \frac{76}{21} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x-30 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} -30 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & -39 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-39}{-1} \\\Leftrightarrow & x = 39 & & \\ & V = \left\{ 39 \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x-\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+15& = & -32x-48 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{+32x} & = & \color{red}{-32x} -48 \color{blue}{+32x} \color{blue}{-15} \\\Leftrightarrow & 44x& = & -63 \\\Leftrightarrow & \frac{44x}{ \color{red}{44} }& = & \frac{-63}{44} \\\Leftrightarrow & x = \frac{-63}{44} & & \\ & V = \left\{ \frac{-63}{44} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{4}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 24x+210 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-24x} & = & \color{red}{24x} +210 \color{blue}{-24x} \color{blue}{+8} \\\Leftrightarrow & -19x& = & 218 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{218}{-19} \\\Leftrightarrow & x = \frac{-218}{19} & & \\ & V = \left\{ \frac{-218}{19} \right\} & \\\end{align}\)
9. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 7x+14 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-7x} & = & \color{red}{7x} +14 \color{blue}{-7x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 18 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{18}{-5} \\\Leftrightarrow & x = \frac{-18}{5} & & \\ & V = \left\{ \frac{-18}{5} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{9}& = & \frac{7}{4}x+6 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{63}{ \color{blue}{36} }x+\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-8& = & 63x+216 \\\Leftrightarrow & 12x \color{red}{-8} \color{blue}{+8} \color{blue}{-63x} & = & \color{red}{63x} +216 \color{blue}{-63x} \color{blue}{+8} \\\Leftrightarrow & -51x& = & 224 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{224}{-51} \\\Leftrightarrow & x = \frac{-224}{51} & & \\ & V = \left\{ \frac{-224}{51} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 7, 10 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{-3}{4}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{20x}{ \color{blue}{140} }- \frac{ 42 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 20x-42& = & -105x-980 \\\Leftrightarrow & 20x \color{red}{-42} \color{blue}{+42} \color{blue}{+105x} & = & \color{red}{-105x} -980 \color{blue}{+105x} \color{blue}{+42} \\\Leftrightarrow & 125x& = & -938 \\\Leftrightarrow & \frac{125x}{ \color{red}{125} }& = & \frac{-938}{125} \\\Leftrightarrow & x = \frac{-938}{125} & & \\ & V = \left\{ \frac{-938}{125} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -112x-252 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+112x} & = & \color{red}{-112x} -252 \color{blue}{+112x} \color{blue}{+12} \\\Leftrightarrow & 133x& = & -240 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-240}{133} \\\Leftrightarrow & x = \frac{-240}{133} & & \\ & V = \left\{ \frac{-240}{133} \right\} & \\\end{align}\)