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. \(\frac{x}{3}-\frac{3}{8}=\frac{1}{2}x+1\)
2. \(\frac{x}{2}-\frac{3}{13}=\frac{-2}{3}x+1\)
3. \(\frac{x}{5}+\frac{5}{6}=\frac{1}{6}x-8\)
4. \(\frac{x}{4}-\frac{3}{10}=\frac{5}{3}x-3\)
5. \(\frac{x}{4}-\frac{4}{7}=\frac{-2}{3}x-7\)
6. \(\frac{x}{2}+\frac{5}{11}=\frac{-5}{3}x+1\)
7. \(\frac{x}{7}-\frac{2}{15}=\frac{-7}{5}x+3\)
8. \(\frac{x}{6}-\frac{2}{9}=\frac{-7}{5}x+8\)
9. \(\frac{x}{3}+\frac{3}{16}=\frac{1}{2}x-5\)
10. \(\frac{x}{7}+\frac{5}{14}=\frac{-3}{4}x+6\)
11. \(\frac{x}{7}-\frac{5}{11}=\frac{3}{5}x-6\)
12. \(\frac{x}{7}-\frac{5}{6}=\frac{1}{3}x-4\)

---

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

Verbetersleutel

1. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x+\frac{24}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 12x+24 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} +24 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & -4x& = & 33 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{33}{-4} \\\Leftrightarrow & x = \frac{-33}{4} & & \\ & V = \left\{ \frac{-33}{4} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & -52x+78 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{+52x} & = & \color{red}{-52x} +78 \color{blue}{+52x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & 96 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{96}{91} \\\Leftrightarrow & x = \frac{96}{91} & & \\ & V = \left\{ \frac{96}{91} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 5x-240 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-5x} & = & \color{red}{5x} -240 \color{blue}{-5x} \color{blue}{-25} \\\Leftrightarrow & x& = & -265 \\ & V = \left\{ -265 \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{5}{3}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{100}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 100x-180 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-100x} & = & \color{red}{100x} -180 \color{blue}{-100x} \color{blue}{+18} \\\Leftrightarrow & -85x& = & -162 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{-162}{-85} \\\Leftrightarrow & x = \frac{162}{85} & & \\ & V = \left\{ \frac{162}{85} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -56x-588 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+56x} & = & \color{red}{-56x} -588 \color{blue}{+56x} \color{blue}{+48} \\\Leftrightarrow & 77x& = & -540 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-540}{77} \\\Leftrightarrow & x = \frac{-540}{77} & & \\ & V = \left\{ \frac{-540}{77} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & -110x+66 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+110x} & = & \color{red}{-110x} +66 \color{blue}{+110x} \color{blue}{-30} \\\Leftrightarrow & 143x& = & 36 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{36}{143} \\\Leftrightarrow & x = \frac{36}{143} & & \\ & V = \left\{ \frac{36}{143} \right\} & \\\end{align}\)
7. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-147}{ \color{blue}{105} }x+\frac{315}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & -147x+315 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{+147x} & = & \color{red}{-147x} +315 \color{blue}{+147x} \color{blue}{+14} \\\Leftrightarrow & 162x& = & 329 \\\Leftrightarrow & \frac{162x}{ \color{red}{162} }& = & \frac{329}{162} \\\Leftrightarrow & x = \frac{329}{162} & & \\ & V = \left\{ \frac{329}{162} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{9}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-126}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-20& = & -126x+720 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{+126x} & = & \color{red}{-126x} +720 \color{blue}{+126x} \color{blue}{+20} \\\Leftrightarrow & 141x& = & 740 \\\Leftrightarrow & \frac{141x}{ \color{red}{141} }& = & \frac{740}{141} \\\Leftrightarrow & x = \frac{740}{141} & & \\ & V = \left\{ \frac{740}{141} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x-240 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} -240 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & -249 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-249}{-8} \\\Leftrightarrow & x = \frac{249}{8} & & \\ & V = \left\{ \frac{249}{8} \right\} & \\\end{align}\)
10. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{-3}{4}x+6 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 10 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x+\frac{168}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+10& = & -21x+168 \\\Leftrightarrow & 4x \color{red}{+10} \color{blue}{-10} \color{blue}{+21x} & = & \color{red}{-21x} +168 \color{blue}{+21x} \color{blue}{-10} \\\Leftrightarrow & 25x& = & 158 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{158}{25} \\\Leftrightarrow & x = \frac{158}{25} & & \\ & V = \left\{ \frac{158}{25} \right\} & \\\end{align}\)
11. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 175 }{ \color{blue}{385} })& = & (\frac{231}{ \color{blue}{385} }x-\frac{2310}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-175& = & 231x-2310 \\\Leftrightarrow & 55x \color{red}{-175} \color{blue}{+175} \color{blue}{-231x} & = & \color{red}{231x} -2310 \color{blue}{-231x} \color{blue}{+175} \\\Leftrightarrow & -176x& = & -2135 \\\Leftrightarrow & \frac{-176x}{ \color{red}{-176} }& = & \frac{-2135}{-176} \\\Leftrightarrow & x = \frac{2135}{176} & & \\ & V = \left\{ \frac{2135}{176} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 14x-168 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-14x} & = & \color{red}{14x} -168 \color{blue}{-14x} \color{blue}{+35} \\\Leftrightarrow & -8x& = & -133 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-133}{-8} \\\Leftrightarrow & x = \frac{133}{8} & & \\ & V = \left\{ \frac{133}{8} \right\} & \\\end{align}\)