Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{-5}{6}x+1 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x+\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & -200x+240 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{+200x} & = & \color{red}{-200x} +240 \color{blue}{+200x} \color{blue}{-75} \\\Leftrightarrow & 248x& = & 165 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{165}{248} \\\Leftrightarrow & x = \frac{165}{248} & & \\ & V = \left\{ \frac{165}{248} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-208}{ \color{blue}{78} }x+\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & -208x+546 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+208x} & = & \color{red}{-208x} +546 \color{blue}{+208x} \color{blue}{-24} \\\Leftrightarrow & 247x& = & 522 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{522}{247} \\\Leftrightarrow & x = \frac{522}{247} & & \\ & V = \left\{ \frac{522}{247} \right\} & \\\end{align}\)
3. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 20 }{ \color{blue}{45} })& = & (\frac{-120}{ \color{blue}{45} }x+\frac{180}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-20& = & -120x+180 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{+120x} & = & \color{red}{-120x} +180 \color{blue}{+120x} \color{blue}{+20} \\\Leftrightarrow & 129x& = & 200 \\\Leftrightarrow & \frac{129x}{ \color{red}{129} }& = & \frac{200}{129} \\\Leftrightarrow & x = \frac{200}{129} & & \\ & V = \left\{ \frac{200}{129} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{7}{2}x+6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{385}{ \color{blue}{110} }x+\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+40& = & 385x+660 \\\Leftrightarrow & 22x \color{red}{+40} \color{blue}{-40} \color{blue}{-385x} & = & \color{red}{385x} +660 \color{blue}{-385x} \color{blue}{-40} \\\Leftrightarrow & -363x& = & 620 \\\Leftrightarrow & \frac{-363x}{ \color{red}{-363} }& = & \frac{620}{-363} \\\Leftrightarrow & x = \frac{-620}{363} & & \\ & V = \left\{ \frac{-620}{363} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{9}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-20& = & 18x-360 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-18x} & = & \color{red}{18x} -360 \color{blue}{-18x} \color{blue}{+20} \\\Leftrightarrow & -3x& = & -340 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-340}{-3} \\\Leftrightarrow & x = \frac{340}{3} & & \\ & V = \left\{ \frac{340}{3} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{6}x-2 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x-\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+120& = & 65x-780 \\\Leftrightarrow & 78x \color{red}{+120} \color{blue}{-120} \color{blue}{-65x} & = & \color{red}{65x} -780 \color{blue}{-65x} \color{blue}{-120} \\\Leftrightarrow & 13x& = & -900 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-900}{13} \\\Leftrightarrow & x = \frac{-900}{13} & & \\ & V = \left\{ \frac{-900}{13} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 28x-252 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-28x} & = & \color{red}{28x} -252 \color{blue}{-28x} \color{blue}{-36} \\\Leftrightarrow & -7x& = & -288 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-288}{-7} \\\Leftrightarrow & x = \frac{288}{7} & & \\ & V = \left\{ \frac{288}{7} \right\} & \\\end{align}\)
8. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }- \frac{ 60 }{ \color{blue}{195} })& = & (\frac{-78}{ \color{blue}{195} }x+\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x-60& = & -78x+585 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{+78x} & = & \color{red}{-78x} +585 \color{blue}{+78x} \color{blue}{+60} \\\Leftrightarrow & 143x& = & 645 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{645}{143} \\\Leftrightarrow & x = \frac{645}{143} & & \\ & V = \left\{ \frac{645}{143} \right\} & \\\end{align}\)
9. \(\text{40 is het kleinste gemene veelvoud van 4, 8 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{8}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{40.} (\frac{10x}{ \color{blue}{40} }- \frac{ 25 }{ \color{blue}{40} })& = & (\frac{-32}{ \color{blue}{40} }x-\frac{40}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 10x-25& = & -32x-40 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{+32x} & = & \color{red}{-32x} -40 \color{blue}{+32x} \color{blue}{+25} \\\Leftrightarrow & 42x& = & -15 \\\Leftrightarrow & \frac{42x}{ \color{red}{42} }& = & \frac{-15}{42} \\\Leftrightarrow & x = \frac{-5}{14} & & \\ & V = \left\{ \frac{-5}{14} \right\} & \\\end{align}\)
10. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{260}{ \color{blue}{195} }x+\frac{390}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+45& = & 260x+390 \\\Leftrightarrow & 39x \color{red}{+45} \color{blue}{-45} \color{blue}{-260x} & = & \color{red}{260x} +390 \color{blue}{-260x} \color{blue}{-45} \\\Leftrightarrow & -221x& = & 345 \\\Leftrightarrow & \frac{-221x}{ \color{red}{-221} }& = & \frac{345}{-221} \\\Leftrightarrow & x = \frac{-345}{221} & & \\ & V = \left\{ \frac{-345}{221} \right\} & \\\end{align}\)
11. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{11}& = & \frac{5}{6}x-2 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{275}{ \color{blue}{330} }x-\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-90& = & 275x-660 \\\Leftrightarrow & 66x \color{red}{-90} \color{blue}{+90} \color{blue}{-275x} & = & \color{red}{275x} -660 \color{blue}{-275x} \color{blue}{+90} \\\Leftrightarrow & -209x& = & -570 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-570}{-209} \\\Leftrightarrow & x = \frac{30}{11} & & \\ & V = \left\{ \frac{30}{11} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 20 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-20& = & 55x-550 \\\Leftrightarrow & 22x \color{red}{-20} \color{blue}{+20} \color{blue}{-55x} & = & \color{red}{55x} -550 \color{blue}{-55x} \color{blue}{+20} \\\Leftrightarrow & -33x& = & -530 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-530}{-33} \\\Leftrightarrow & x = \frac{530}{33} & & \\ & V = \left\{ \frac{530}{33} \right\} & \\\end{align}\)