Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x-\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & 22x-198 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{-22x} & = & \color{red}{22x} -198 \color{blue}{-22x} \color{blue}{-12} \\\Leftrightarrow & 11x& = & -210 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-210}{11} \\\Leftrightarrow & x = \frac{-210}{11} & & \\ & V = \left\{ \frac{-210}{11} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & -56x-490 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+56x} & = & \color{red}{-56x} -490 \color{blue}{+56x} \color{blue}{-40} \\\Leftrightarrow & 91x& = & -530 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-530}{91} \\\Leftrightarrow & x = \frac{-530}{91} & & \\ & V = \left\{ \frac{-530}{91} \right\} & \\\end{align}\)
3. \(\text{15 is het kleinste gemene veelvoud van 5, 15 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{15.} (\frac{3x}{ \color{blue}{15} }+ \frac{ 2 }{ \color{blue}{15} })& = & (\frac{-40}{ \color{blue}{15} }x+\frac{15}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 3x+2& = & -40x+15 \\\Leftrightarrow & 3x \color{red}{+2} \color{blue}{-2} \color{blue}{+40x} & = & \color{red}{-40x} +15 \color{blue}{+40x} \color{blue}{-2} \\\Leftrightarrow & 43x& = & 13 \\\Leftrightarrow & \frac{43x}{ \color{red}{43} }& = & \frac{13}{43} \\\Leftrightarrow & x = \frac{13}{43} & & \\ & V = \left\{ \frac{13}{43} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 168x+420 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-168x} & = & \color{red}{168x} +420 \color{blue}{-168x} \color{blue}{-80} \\\Leftrightarrow & -133x& = & 340 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{340}{-133} \\\Leftrightarrow & x = \frac{-340}{133} & & \\ & V = \left\{ \frac{-340}{133} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{7}{2}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 147x-336 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-147x} & = & \color{red}{147x} -336 \color{blue}{-147x} \color{blue}{-24} \\\Leftrightarrow & -133x& = & -360 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-360}{-133} \\\Leftrightarrow & x = \frac{360}{133} & & \\ & V = \left\{ \frac{360}{133} \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-24& = & 44x+792 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{-44x} & = & \color{red}{44x} +792 \color{blue}{-44x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & 816 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{816}{-11} \\\Leftrightarrow & x = \frac{-816}{11} & & \\ & V = \left\{ \frac{-816}{11} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & 18x-720 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{-18x} & = & \color{red}{18x} -720 \color{blue}{-18x} \color{blue}{+50} \\\Leftrightarrow & -3x& = & -670 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-670}{-3} \\\Leftrightarrow & x = \frac{670}{3} & & \\ & V = \left\{ \frac{670}{3} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x+\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & -112x+280 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+112x} & = & \color{red}{-112x} +280 \color{blue}{+112x} \color{blue}{+40} \\\Leftrightarrow & 147x& = & 320 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{320}{147} \\\Leftrightarrow & x = \frac{320}{147} & & \\ & V = \left\{ \frac{320}{147} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & -312x-780 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{+312x} & = & \color{red}{-312x} -780 \color{blue}{+312x} \color{blue}{+120} \\\Leftrightarrow & 377x& = & -660 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-660}{377} \\\Leftrightarrow & x = \frac{-660}{377} & & \\ & V = \left\{ \frac{-660}{377} \right\} & \\\end{align}\)
10. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }+ \frac{ 112 }{ \color{blue}{252} })& = & (\frac{63}{ \color{blue}{252} }x-\frac{1512}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x+112& = & 63x-1512 \\\Leftrightarrow & 36x \color{red}{+112} \color{blue}{-112} \color{blue}{-63x} & = & \color{red}{63x} -1512 \color{blue}{-63x} \color{blue}{-112} \\\Leftrightarrow & -27x& = & -1624 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-1624}{-27} \\\Leftrightarrow & x = \frac{1624}{27} & & \\ & V = \left\{ \frac{1624}{27} \right\} & \\\end{align}\)
11. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{11}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 30 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-30& = & 55x-660 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-55x} & = & \color{red}{55x} -660 \color{blue}{-55x} \color{blue}{+30} \\\Leftrightarrow & -33x& = & -630 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-630}{-33} \\\Leftrightarrow & x = \frac{210}{11} & & \\ & V = \left\{ \frac{210}{11} \right\} & \\\end{align}\)
12. \(\text{168 is het kleinste gemene veelvoud van 7, 8 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{168.} (\frac{24x}{ \color{blue}{168} }+ \frac{ 63 }{ \color{blue}{168} })& = & (\frac{28}{ \color{blue}{168} }x+\frac{168}{ \color{blue}{168} }) \color{blue}{.168} \\\Leftrightarrow & 24x+63& = & 28x+168 \\\Leftrightarrow & 24x \color{red}{+63} \color{blue}{-63} \color{blue}{-28x} & = & \color{red}{28x} +168 \color{blue}{-28x} \color{blue}{-63} \\\Leftrightarrow & -4x& = & 105 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{105}{-4} \\\Leftrightarrow & x = \frac{-105}{4} & & \\ & V = \left\{ \frac{-105}{4} \right\} & \\\end{align}\)