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Solution 2.2:2a

From Förberedande kurs i matematik 1

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m (Lösning 2.2:2a moved to Solution 2.2:2a: Robot: moved page)
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If we divide up the denominators that appear in the equation into small integer factors
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<center> [[Image:2_2_2a-1(2).gif]] </center>
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<math>6=2\centerdot 3</math>,
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<math>9=3\centerdot 3</math>
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and 2, we see that the lowest common denominator is
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<center> [[Image:2_2_2a-2(2).gif]] </center>
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<math>2\centerdot 3\centerdot 3=18</math>. Thus, we multiply both sides of the equation by
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<math>2\centerdot 3\centerdot 3</math>
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in order to avoid having denominators in the equation:
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<math>\begin{align}
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& 2\centerdot 3\centerdot 3\centerdot \frac{5x}{6}-2\centerdot 3\centerdot 3\centerdot \frac{x+2}{9}=2\centerdot 3\centerdot 3\centerdot \frac{1}{2} \\
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& \Leftrightarrow 3\centerdot 5x-2\centerdot \left( x+2 \right)=3\centerdot 3 \\
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\end{align}</math>
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We can rewrite the left-hand side as
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<math>3\centerdot 5x-2\centerdot \left( x+2 \right)=15x-2x-4=13x-4</math>, so that we get the equation
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<math>13x-4=9</math>
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We can now solve this first-degree equation by carrying out simple arithmetical calculations so as to get
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<math>x</math>
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by itself on one side:
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1. Add 4 to both sides:
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<math>13x-+4=9+4</math>
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which gives
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<math>13x=13</math>
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.
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2. Divide both sides by 13:
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<math>\frac{13x}{13}=\frac{13}{13}</math>
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which gives the answer
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<math>x=1</math>.
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The equation has
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<math>1</math>
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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
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<math>x=1</math>
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really is the correct answer( i.e. that we haven't calculated incorrectly):
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LHS =
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<math>\frac{5\centerdot 1}{6}-\frac{1\centerdot 2}{9}=\frac{5}{6}-\frac{3}{9}=\frac{5}{6}-\frac{1}{3}=\frac{5}{6}-\frac{1\centerdot 2}{3\centerdot 2}=\frac{5-2}{6}=\frac{3}{6}=\frac{1}{2}</math>
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= RHS

Revision as of 12:25, 17 September 2008

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−2x+2=33


We can rewrite the left-hand side as 35x−2x+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=9+4

which gives

13x=13 .

2. Divide both sides by 13:

1313x=1313

which gives the answer x=1.

The equation has 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 = 651−912=65−93=65−31=65−3212=65−2=63=21 = RHS

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