Lösung 2.2:2b
Aus Online Mathematik Brückenkurs 1
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| - | First, we multiply both sides in the equation by | + | First, we multiply both sides in the equation by <math>4\cdot 7=28</math>, so that we get rid of the denominators in the equation, |
| - | <math>4\ | + | |
| + | {{Displayed math||<math>\begin{align} | ||
| + | & 4\cdot{}\rlap{/}7\cdot\frac{8x+3}{\rlap{/}7} - \rlap{/}4\cdot 7\cdot\frac{5x-7}{\rlap{/}4} = 4\cdot 7\cdot 2\\[5pt] | ||
| + | &\qquad\Leftrightarrow\quad 4\cdot (8x+3) - 7\cdot (5x-7) = 56\,\textrm{.} | ||
| + | \end{align}</math>}} | ||
| - | <math> | + | We can simplify the left-hand side to <math>4\cdot (8x+3) - 7\cdot (5x-7) = 32x+12-35x+49 = -3x+61\,</math>. Hence, the equation is |
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| + | {{Displayed math||<math>-3x+61=56\,\textrm{.}</math>}} | ||
| - | We | + | We solve this equation by subtracting 61 from both sides and then dividing by -3, |
| + | {{Displayed math||<math>\begin{align} | ||
| + | -3x+61-61&=56-61\,,\\[5pt] | ||
| + | -3x&=-5\,,\\[5pt] | ||
| + | \frac{-3x}{-3}&=\frac{-5}{-3}\,,\\[5pt] | ||
| + | x&=\frac{5}{3}\,\textrm{.} | ||
| + | \end{align}</math>}} | ||
| - | <math> | + | The answer is <math>x={5}/{3}\,</math>. |
| + | As the final part of the solution, check the answer by substituting <math>x={5}/{3}</math> into the original equation | ||
| - | + | {{Displayed math||<math>\begin{align} | |
| - | + | \text{LHS} | |
| - | + | &= \frac{8\cdot\frac{5}{3}+3}{7}-\frac{5\cdot\frac{5}{3}-7}{4} | |
| - | + | = \frac{\bigl(8\cdot\frac{5}{3}+3\bigr)\cdot 3}{7\cdot 3} - \frac{\bigl( 5\cdot \frac{5}{3}-7\bigr)\cdot 3}{4\cdot 3}\\[5pt] | |
| - | + | &= \frac{8\cdot 5+3\cdot 3}{7\cdot 3}-\frac{5\cdot 5-7\cdot 3}{4\cdot 3} | |
| - | + | = \frac{40+9}{21}-\frac{25-21}{12}\\[5pt] | |
| - | + | &= \frac{49}{21}-\frac{4}{12} | |
| - | + | = \frac{7\cdot 7}{3\cdot 7} - \frac{2\cdot 2}{2\cdot 2\cdot 3} | |
| - | + | = \frac{7}{3}-\frac{1}{3} | |
| - | + | = \frac{7-1}{3} | |
| - | + | = \frac{6}{3} = 2 = \text{RHS.} | |
| - | + | \end{align}</math>}} | |
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| - | <math>\begin{align} | + | |
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| - | & =\frac{8\ | + | |
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| - | & =\frac{49}{21}-\frac{4}{12}=\frac{7\ | + | |
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| - | \end{align}</math> | + | |
Version vom 14:05, 23. Sep. 2008
First, we multiply both sides in the equation by \displaystyle 4\cdot 7=28, so that we get rid of the denominators in the equation,
We can simplify the left-hand side to \displaystyle 4\cdot (8x+3) - 7\cdot (5x-7) = 32x+12-35x+49 = -3x+61\,. Hence, the equation is
We solve this equation by subtracting 61 from both sides and then dividing by -3,
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
