Lösung 2.2:2b

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First, we multiply both sides in the equation by
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<center> [[Image:2_2_2b-1(2).gif]] </center>
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<math>4\centerdot 7=28</math>, so that we get rid of the denominators in the equation,
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<center> [[Image:2_2_2b-2(2).gif]] </center>
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<math>\begin{align}
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& 4\centerdot 7\centerdot \frac{8x+3}{7}-4\centerdot 7\centerdot \frac{5x-7}{4}=4\centerdot 7\centerdot 2 \\
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& \Leftrightarrow 4\centerdot \left( 8x+3 \right)-7\centerdot \left( 5x-7 \right)=56 \\
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\end{align}</math>
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We can simplify the left-hand side to ,
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<math>4\centerdot \left( 8x+3 \right)-7\centerdot \left( 5x-7 \right)=32x+12-35x+49=-3x+61</math>
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Hence, the equation is
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<math>-3x+61=56</math>
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We solve this equation by subtracting
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<math>61</math>
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from both sides and then dividing by
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<math>-3</math>,
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<math>\begin{align}
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& -3x+61-61=56-61 \\
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& -3x=-5 \\
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& \frac{-3x}{-3}=\frac{-5}{-3} \\
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& x=\frac{5}{3} \\
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\end{align}</math>
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The answer is
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<math>x={5}/{3}\;</math>.
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As the final part of the solution, check the answer by substituting
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<math>x={5}/{3}\;</math>
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into the original equation
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<math>\begin{align}
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& \text{LHS}\quad =\quad \frac{8\centerdot \frac{5}{3}+3}{7}-\frac{5\centerdot \frac{5}{3}-7}{4}=\frac{\left( 8\centerdot \frac{5}{3}+3 \right)\centerdot 3}{7\centerdot 3}-\frac{\left( 5\centerdot \frac{5}{3}-7 \right)\centerdot 3}{4\centerdot 3} \\
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& \\
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& =\frac{8\centerdot 5+3\centerdot 3}{7\centerdot 3}-\frac{5\centerdot 5-7\centerdot 3}{4\centerdot 3}=\frac{40+9}{21}-\frac{25-21}{12} \\
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& \\
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& =\frac{49}{21}-\frac{4}{12}=\frac{7\centerdot 7}{3\centerdot 7}-\frac{2\centerdot 2}{2\centerdot 2\centerdot 2}=\frac{7}{3}-\frac{1}{3}=\frac{7-1}{3} \\
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& \\
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& =\frac{6}{3}=2\quad =\quad \text{RHS} \\
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\end{align}</math>

Version vom 12:45, 17. Sep. 2008

First, we multiply both sides in the equation by \displaystyle 4\centerdot 7=28, so that we get rid of the denominators in the equation,


\displaystyle \begin{align} & 4\centerdot 7\centerdot \frac{8x+3}{7}-4\centerdot 7\centerdot \frac{5x-7}{4}=4\centerdot 7\centerdot 2 \\ & \Leftrightarrow 4\centerdot \left( 8x+3 \right)-7\centerdot \left( 5x-7 \right)=56 \\ \end{align}


We can simplify the left-hand side to ,


\displaystyle 4\centerdot \left( 8x+3 \right)-7\centerdot \left( 5x-7 \right)=32x+12-35x+49=-3x+61


Hence, the equation is


\displaystyle -3x+61=56


We solve this equation by subtracting \displaystyle 61 from both sides and then dividing by \displaystyle -3,


\displaystyle \begin{align} & -3x+61-61=56-61 \\ & -3x=-5 \\ & \frac{-3x}{-3}=\frac{-5}{-3} \\ & x=\frac{5}{3} \\ \end{align}


The answer is \displaystyle x={5}/{3}\;.

As the final part of the solution, check the answer by substituting \displaystyle x={5}/{3}\; into the original equation


\displaystyle \begin{align} & \text{LHS}\quad =\quad \frac{8\centerdot \frac{5}{3}+3}{7}-\frac{5\centerdot \frac{5}{3}-7}{4}=\frac{\left( 8\centerdot \frac{5}{3}+3 \right)\centerdot 3}{7\centerdot 3}-\frac{\left( 5\centerdot \frac{5}{3}-7 \right)\centerdot 3}{4\centerdot 3} \\ & \\ & =\frac{8\centerdot 5+3\centerdot 3}{7\centerdot 3}-\frac{5\centerdot 5-7\centerdot 3}{4\centerdot 3}=\frac{40+9}{21}-\frac{25-21}{12} \\ & \\ & =\frac{49}{21}-\frac{4}{12}=\frac{7\centerdot 7}{3\centerdot 7}-\frac{2\centerdot 2}{2\centerdot 2\centerdot 2}=\frac{7}{3}-\frac{1}{3}=\frac{7-1}{3} \\ & \\ & =\frac{6}{3}=2\quad =\quad \text{RHS} \\ \end{align}

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_2.2:2b“