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

---

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & 78x-780 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{-78x} & = & \color{red}{78x} -780 \color{blue}{-78x} \color{blue}{-150} \\\Leftrightarrow & -13x& = & -930 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-930}{-13} \\\Leftrightarrow & x = \frac{930}{13} & & \\ & V = \left\{ \frac{930}{13} \right\} & \\\end{align}\)
2. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{-208}{ \color{blue}{260} }x-\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-60& = & -208x-2080 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{+208x} & = & \color{red}{-208x} -2080 \color{blue}{+208x} \color{blue}{+60} \\\Leftrightarrow & 273x& = & -2020 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{-2020}{273} \\\Leftrightarrow & x = \frac{-2020}{273} & & \\ & V = \left\{ \frac{-2020}{273} \right\} & \\\end{align}\)
3. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{-3}{4}x+6 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{-195}{ \color{blue}{260} }x+\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x+40& = & -195x+1560 \\\Leftrightarrow & 52x \color{red}{+40} \color{blue}{-40} \color{blue}{+195x} & = & \color{red}{-195x} +1560 \color{blue}{+195x} \color{blue}{-40} \\\Leftrightarrow & 247x& = & 1520 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{1520}{247} \\\Leftrightarrow & x = \frac{80}{13} & & \\ & V = \left\{ \frac{80}{13} \right\} & \\\end{align}\)
4. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{7}{4}x+5 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 16 }{ \color{blue}{28} })& = & (\frac{49}{ \color{blue}{28} }x+\frac{140}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+16& = & 49x+140 \\\Leftrightarrow & 4x \color{red}{+16} \color{blue}{-16} \color{blue}{-49x} & = & \color{red}{49x} +140 \color{blue}{-49x} \color{blue}{-16} \\\Leftrightarrow & -45x& = & 124 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{124}{-45} \\\Leftrightarrow & x = \frac{-124}{45} & & \\ & V = \left\{ \frac{-124}{45} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{-3}{4}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+24& = & -63x+420 \\\Leftrightarrow & 28x \color{red}{+24} \color{blue}{-24} \color{blue}{+63x} & = & \color{red}{-63x} +420 \color{blue}{+63x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & 396 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{396}{91} \\\Leftrightarrow & x = \frac{396}{91} & & \\ & V = \left\{ \frac{396}{91} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{364}{ \color{blue}{156} }x-\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 364x-1248 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-364x} & = & \color{red}{364x} -1248 \color{blue}{-364x} \color{blue}{-48} \\\Leftrightarrow & -325x& = & -1296 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{-1296}{-325} \\\Leftrightarrow & x = \frac{1296}{325} & & \\ & V = \left\{ \frac{1296}{325} \right\} & \\\end{align}\)
7. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x+\frac{48}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x+5& = & 3x+48 \\\Leftrightarrow & 2x \color{red}{+5} \color{blue}{-5} \color{blue}{-3x} & = & \color{red}{3x} +48 \color{blue}{-3x} \color{blue}{-5} \\\Leftrightarrow & -x& = & 43 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{43}{-1} \\\Leftrightarrow & x = -43 & & \\ & V = \left\{ -43 \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 168x+420 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-168x} & = & \color{red}{168x} +420 \color{blue}{-168x} \color{blue}{-90} \\\Leftrightarrow & -133x& = & 330 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{330}{-133} \\\Leftrightarrow & x = \frac{-330}{133} & & \\ & V = \left\{ \frac{-330}{133} \right\} & \\\end{align}\)
9. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 24x+192 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-24x} & = & \color{red}{24x} +192 \color{blue}{-24x} \color{blue}{+9} \\\Leftrightarrow & -8x& = & 201 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{201}{-8} \\\Leftrightarrow & x = \frac{-201}{8} & & \\ & V = \left\{ \frac{-201}{8} \right\} & \\\end{align}\)
10. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{2}{3}x-6 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{70}{ \color{blue}{105} }x-\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & 70x-630 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{-70x} & = & \color{red}{70x} -630 \color{blue}{-70x} \color{blue}{+14} \\\Leftrightarrow & -55x& = & -616 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-616}{-55} \\\Leftrightarrow & x = \frac{56}{5} & & \\ & V = \left\{ \frac{56}{5} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & 168x-700 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-168x} & = & \color{red}{168x} -700 \color{blue}{-168x} \color{blue}{+40} \\\Leftrightarrow & -133x& = & -660 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-660}{-133} \\\Leftrightarrow & x = \frac{660}{133} & & \\ & V = \left\{ \frac{660}{133} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{3}{4}x-6 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 36 }{ \color{blue}{132} })& = & (\frac{99}{ \color{blue}{132} }x-\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-36& = & 99x-792 \\\Leftrightarrow & 44x \color{red}{-36} \color{blue}{+36} \color{blue}{-99x} & = & \color{red}{99x} -792 \color{blue}{-99x} \color{blue}{+36} \\\Leftrightarrow & -55x& = & -756 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-756}{-55} \\\Leftrightarrow & x = \frac{756}{55} & & \\ & V = \left\{ \frac{756}{55} \right\} & \\\end{align}\)