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Solution 2.2:2a

From Förberedande kurs i matematik 1

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Current revision (13:44, 23 September 2008) (edit) (undo)
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If we divide up the denominators that appear in the equation into small integer factors <math>6=2\cdot 3</math>, <math>9=3\cdot 3</math> and 2, we see that the lowest common denominator is <math>2\cdot 3\cdot 3=18</math>. Thus, we multiply both sides of the equation by <math>2\cdot 3\cdot 3</math> in order to avoid having denominators in the equation
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<center> [[Bild:2_2_2a-1(2).gif]] </center>
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{{Displayed math||<math>\begin{align}
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& \rlap{/}2\cdot{}\rlap{/}3\cdot 3\cdot\frac{5x}{\rlap{/}6} - 2\cdot{}\rlap{/}3\cdot{}\rlap{/}3\cdot\frac{x+2}{\rlap{/}9} = \rlap{/}2\cdot 3\cdot 3\cdot \frac{1}{\rlap{/}2} \\[5pt]
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&\qquad\Leftrightarrow\quad 3\cdot 5x-2\cdot (x+2) = 3\cdot 3\,\textrm{.}\\
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\end{align}</math>}}
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We can rewrite the left-hand side as <math>3\cdot 5x-2\cdot (x+2) = 15x-2x-4 = 13x-4</math>, so that we get the equation
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{{Displayed math||<math>13x-4=9\,\textrm{.}</math>}}
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We can now solve this first-degree equation by carrying out simple arithmetical calculations so as to get ''x'' by itself on one side:
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<ol>
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<li>Add 4 to both sides, <math>\vphantom{x_2}13x-4+4=9+4\,,</math> which gives <math>\ 13x=13\,\textrm{.}</math></li>
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<li>Divide both sides by 13, <math>\frac{13x}{13}=\frac{13}{13}\,,</math> which gives the answer <math>\ x=1\,\textrm{.}</math></li>
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</ol>
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The equation has <math>x=1</math> as the solution.
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When we have obtained an answer, it is important to go back to the original equation to check that <math>x=1</math> really is the correct answer (i.e. that we haven't calculated incorrectly)
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{{Displayed math||<math>\begin{align}
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\text{LHS}
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&= \frac{5\cdot 1}{6}-\frac{1+2}{9} = \frac{5}{6}-\frac{3}{9} = \frac{5}{6}-\frac{1}{3}\\[5pt]
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&= \frac{5}{6}-\frac{1\cdot 2}{3\cdot 2} = \frac{5-2}{6} = \frac{3}{6} = \frac{1}{2}
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= \text{RHS}\,\textrm{.}
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\end{align}</math>}}

Current revision

If we divide up the denominators that appear in the equation into small integer factors 6=23, 9=33 and 2, we see that the lowest common denominator is 233=18. Thus, we multiply both sides of the equation by 233 in order to avoid having denominators in the equation

23365x−2339x+2=2332135x−2(x+2)=33.

We can rewrite the left-hand side as 35x−2(x+2)=15x−2x−4=13x−4, so that we get the equation

13x−4=9.

We can now solve this first-degree equation by carrying out simple arithmetical calculations so as to get x by itself on one side:

  1. Add 4 to both sides, 13x−4+4=9+4 which gives  13x=13.
  2. Divide both sides by 13, 1313x=1313 which gives the answer  x=1.

The equation has x=1 as the solution.

When we have obtained an answer, it is important to go back to the original equation to check that x=1 really is the correct answer (i.e. that we haven't calculated incorrectly)

LHS=651−91+2=65−93=65−31=65−3212=65−2=63=21=RHS.
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