Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x-150 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} -150 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & -159 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-159}{-1} \\\Leftrightarrow & x = 159 & & \\ & V = \left\{ 159 \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{2}{5}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{24}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 24x-180 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-24x} & = & \color{red}{24x} -180 \color{blue}{-24x} \color{blue}{-8} \\\Leftrightarrow & -9x& = & -188 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-188}{-9} \\\Leftrightarrow & x = \frac{188}{9} & & \\ & V = \left\{ \frac{188}{9} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & 84x+70 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{-84x} & = & \color{red}{84x} +70 \color{blue}{-84x} \color{blue}{-20} \\\Leftrightarrow & -49x& = & 50 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{50}{-49} \\\Leftrightarrow & x = \frac{-50}{49} & & \\ & V = \left\{ \frac{-50}{49} \right\} & \\\end{align}\)
4. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{13}& = & \frac{5}{3}x+4 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 105 }{ \color{blue}{273} })& = & (\frac{455}{ \color{blue}{273} }x+\frac{1092}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-105& = & 455x+1092 \\\Leftrightarrow & 39x \color{red}{-105} \color{blue}{+105} \color{blue}{-455x} & = & \color{red}{455x} +1092 \color{blue}{-455x} \color{blue}{+105} \\\Leftrightarrow & -416x& = & 1197 \\\Leftrightarrow & \frac{-416x}{ \color{red}{-416} }& = & \frac{1197}{-416} \\\Leftrightarrow & x = \frac{-1197}{416} & & \\ & V = \left\{ \frac{-1197}{416} \right\} & \\\end{align}\)
5. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{5}{2}x+4 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 56 }{ \color{blue}{154} })& = & (\frac{385}{ \color{blue}{154} }x+\frac{616}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-56& = & 385x+616 \\\Leftrightarrow & 22x \color{red}{-56} \color{blue}{+56} \color{blue}{-385x} & = & \color{red}{385x} +616 \color{blue}{-385x} \color{blue}{+56} \\\Leftrightarrow & -363x& = & 672 \\\Leftrightarrow & \frac{-363x}{ \color{red}{-363} }& = & \frac{672}{-363} \\\Leftrightarrow & x = \frac{-224}{121} & & \\ & V = \left\{ \frac{-224}{121} \right\} & \\\end{align}\)
6. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{11}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 30 }{ \color{blue}{165} })& = & (\frac{-440}{ \color{blue}{165} }x-\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+30& = & -440x-330 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+440x} & = & \color{red}{-440x} -330 \color{blue}{+440x} \color{blue}{-30} \\\Leftrightarrow & 473x& = & -360 \\\Leftrightarrow & \frac{473x}{ \color{red}{473} }& = & \frac{-360}{473} \\\Leftrightarrow & x = \frac{-360}{473} & & \\ & V = \left\{ \frac{-360}{473} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x-\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & 154x-528 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{-154x} & = & \color{red}{154x} -528 \color{blue}{-154x} \color{blue}{-30} \\\Leftrightarrow & -121x& = & -558 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-558}{-121} \\\Leftrightarrow & x = \frac{558}{121} & & \\ & V = \left\{ \frac{558}{121} \right\} & \\\end{align}\)
8. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{11}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 42 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{770}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+42& = & 77x+770 \\\Leftrightarrow & 22x \color{red}{+42} \color{blue}{-42} \color{blue}{-77x} & = & \color{red}{77x} +770 \color{blue}{-77x} \color{blue}{-42} \\\Leftrightarrow & -55x& = & 728 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{728}{-55} \\\Leftrightarrow & x = \frac{-728}{55} & & \\ & V = \left\{ \frac{-728}{55} \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{2}{3}x-8 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{24}{ \color{blue}{36} }x-\frac{288}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & 24x-288 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{-24x} & = & \color{red}{24x} -288 \color{blue}{-24x} \color{blue}{+16} \\\Leftrightarrow & -15x& = & -272 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-272}{-15} \\\Leftrightarrow & x = \frac{272}{15} & & \\ & V = \left\{ \frac{272}{15} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x-\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 112x-384 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-112x} & = & \color{red}{112x} -384 \color{blue}{-112x} \color{blue}{-15} \\\Leftrightarrow & -88x& = & -399 \\\Leftrightarrow & \frac{-88x}{ \color{red}{-88} }& = & \frac{-399}{-88} \\\Leftrightarrow & x = \frac{399}{88} & & \\ & V = \left\{ \frac{399}{88} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x-1470 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} -1470 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & -1410 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1410}{203} \\\Leftrightarrow & x = \frac{-1410}{203} & & \\ & V = \left\{ \frac{-1410}{203} \right\} & \\\end{align}\)
12. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{-7}{4}x-2 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }- \frac{ 112 }{ \color{blue}{252} })& = & (\frac{-441}{ \color{blue}{252} }x-\frac{504}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x-112& = & -441x-504 \\\Leftrightarrow & 36x \color{red}{-112} \color{blue}{+112} \color{blue}{+441x} & = & \color{red}{-441x} -504 \color{blue}{+441x} \color{blue}{+112} \\\Leftrightarrow & 477x& = & -392 \\\Leftrightarrow & \frac{477x}{ \color{red}{477} }& = & \frac{-392}{477} \\\Leftrightarrow & x = \frac{-392}{477} & & \\ & V = \left\{ \frac{-392}{477} \right\} & \\\end{align}\)