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

---

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -72x-720 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+72x} & = & \color{red}{-72x} -720 \color{blue}{+72x} \color{blue}{-40} \\\Leftrightarrow & 87x& = & -760 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{-760}{87} \\\Leftrightarrow & x = \frac{-760}{87} & & \\ & V = \left\{ \frac{-760}{87} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -140x+588 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+140x} & = & \color{red}{-140x} +588 \color{blue}{+140x} \color{blue}{+24} \\\Leftrightarrow & 161x& = & 612 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{612}{161} \\\Leftrightarrow & x = \frac{612}{161} & & \\ & V = \left\{ \frac{612}{161} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{9}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-20& = & 108x+720 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-108x} & = & \color{red}{108x} +720 \color{blue}{-108x} \color{blue}{+20} \\\Leftrightarrow & -93x& = & 740 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{740}{-93} \\\Leftrightarrow & x = \frac{-740}{93} & & \\ & V = \left\{ \frac{-740}{93} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & 42x+180 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{-42x} & = & \color{red}{42x} +180 \color{blue}{-42x} \color{blue}{+9} \\\Leftrightarrow & -37x& = & 189 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{189}{-37} \\\Leftrightarrow & x = \frac{-189}{37} & & \\ & V = \left\{ \frac{-189}{37} \right\} & \\\end{align}\)
5. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{9}& = & \frac{1}{3}x-8 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 20 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x-\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+20& = & 12x-288 \\\Leftrightarrow & 9x \color{red}{+20} \color{blue}{-20} \color{blue}{-12x} & = & \color{red}{12x} -288 \color{blue}{-12x} \color{blue}{-20} \\\Leftrightarrow & -3x& = & -308 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-308}{-3} \\\Leftrightarrow & x = \frac{308}{3} & & \\ & V = \left\{ \frac{308}{3} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x-\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & 288x-1200 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-288x} & = & \color{red}{288x} -1200 \color{blue}{-288x} \color{blue}{-45} \\\Leftrightarrow & -248x& = & -1245 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{-1245}{-248} \\\Leftrightarrow & x = \frac{1245}{248} & & \\ & V = \left\{ \frac{1245}{248} \right\} & \\\end{align}\)
7. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{55}{ \color{blue}{165} }x-\frac{990}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+75& = & 55x-990 \\\Leftrightarrow & 33x \color{red}{+75} \color{blue}{-75} \color{blue}{-55x} & = & \color{red}{55x} -990 \color{blue}{-55x} \color{blue}{-75} \\\Leftrightarrow & -22x& = & -1065 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{-1065}{-22} \\\Leftrightarrow & x = \frac{1065}{22} & & \\ & V = \left\{ \frac{1065}{22} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{12}& = & \frac{-5}{6}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 35 }{ \color{blue}{84} })& = & (\frac{-70}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+35& = & -70x-252 \\\Leftrightarrow & 12x \color{red}{+35} \color{blue}{-35} \color{blue}{+70x} & = & \color{red}{-70x} -252 \color{blue}{+70x} \color{blue}{-35} \\\Leftrightarrow & 82x& = & -287 \\\Leftrightarrow & \frac{82x}{ \color{red}{82} }& = & \frac{-287}{82} \\\Leftrightarrow & x = \frac{-7}{2} & & \\ & V = \left\{ \frac{-7}{2} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{24}{ \color{blue}{120} }x-\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & 24x-360 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{-24x} & = & \color{red}{24x} -360 \color{blue}{-24x} \color{blue}{-75} \\\Leftrightarrow & -4x& = & -435 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-435}{-4} \\\Leftrightarrow & x = \frac{435}{4} & & \\ & V = \left\{ \frac{435}{4} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-20& = & 35x+70 \\\Leftrightarrow & 14x \color{red}{-20} \color{blue}{+20} \color{blue}{-35x} & = & \color{red}{35x} +70 \color{blue}{-35x} \color{blue}{+20} \\\Leftrightarrow & -21x& = & 90 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{90}{-21} \\\Leftrightarrow & x = \frac{-30}{7} & & \\ & V = \left\{ \frac{-30}{7} \right\} & \\\end{align}\)
11. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x+\frac{42}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 35x+42 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-35x} & = & \color{red}{35x} +42 \color{blue}{-35x} \color{blue}{+4} \\\Leftrightarrow & -33x& = & 46 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{46}{-33} \\\Leftrightarrow & x = \frac{-46}{33} & & \\ & V = \left\{ \frac{-46}{33} \right\} & \\\end{align}\)
12. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 175 }{ \color{blue}{385} })& = & (\frac{-154}{ \color{blue}{385} }x-\frac{2310}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+175& = & -154x-2310 \\\Leftrightarrow & 55x \color{red}{+175} \color{blue}{-175} \color{blue}{+154x} & = & \color{red}{-154x} -2310 \color{blue}{+154x} \color{blue}{-175} \\\Leftrightarrow & 209x& = & -2485 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-2485}{209} \\\Leftrightarrow & x = \frac{-2485}{209} & & \\ & V = \left\{ \frac{-2485}{209} \right\} & \\\end{align}\)