algerba 1
- Thread starter mrs rosa
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2/3 (p - 6) + 4 = 1/6
We cannot solve for a variable, as long as it appears inside parentheses, so the first step is to multiply those factors on the lefthand side.
To multiply the expression (p - 6) by the Rational number 2/3, we apply the Distributive Property:
(2/3)(p) + (2/3)(-6) + 4 = 1/6
Now multiply (2/3)(-6):
2/3 p + (-12/3) + 4 = 1/6
Reduce the Rational number -12/3 to lowest terms:
2/3 p + (-4) + 4 = 1/6
-4 and 4 are opposites; they add up to zero:
2/3 p = 1/6
So far, this tells us what 2/3 p equals, but we're not interested in 2/3rds of p; we want to know what p equals.
In other words, we want the coefficient in front of p to be 1 instead of 2/3.
Multiply both sides of the equation by 3/2, which is the reciprocal of 2/3:
(3/2)(2/3) p = (3/2)(1/6)
Multiplying two reciprocals always gives 1:
(1) p = 3/12
Reduce the Rational number 3/12 to lowest terms:
p = 1/4
We're not quite done, yet:
Check the solution by substituting it for p in the original equation:
2/3 (p - 6) + 4 = 1/6
2/3 (1/4 - 6) + 4 = 1/6
2/3 (1/4 - 24/4) + 4 = 1/6
2/3 (-23/4) + 4 = 1/6
-46/12 + 4 = 1/6
-23/6 + 24/6 = 1/6
1/6 = 1/6
The answer checks.
mrs rosa said:
You have the equivalent of
\(\displaystyle \frac{2}{3}(p - 6) + 4 = \frac{1}{6}\)
I recommend that you clear denominators at the outset. That way, in this certain problem, you will just work
with integer coefficients and integer constants until the end portion of solving for the variable.
The least common denominator, which is the least common multiple of the denominators \(\displaystyle 3 \\) and \(\displaystyle \ 6, \\) is \(\displaystyle \ 6.\)
Multiply each term on both sides by \(\displaystyle 6,\) or by the equivalent form \(\displaystyle \frac{6}{1}\), to clear denominators.
\(\displaystyle \frac{6}{1}[\frac{2}{3}(p - 6)] + 6(4) = \frac{6}{1}(\frac{1}{6})\)
\(\displaystyle 4(p - 6) + 24 = 1\)
\(\displaystyle 4p - 24 + 24 = 1\)
\(\displaystyle 4p = 1\)
\(\displaystyle \frac{4p}{4} = \frac{1}{4}\)
\(\displaystyle \boxed{p = \frac{1}{4}}\)