Answer
The solution to this equation is $t = 2$ and $P = \frac{19}{3}$.
Let's check the answer by substituting both values into one of the original equations to see if both sides are equal:
$2 + 3(\frac{19}{3}) = 21$
Multiply first:
$2 + 19 = 21$
Add:
$21 = 21$
The two sides are equal; therefore, this solution is correct.
This system of equations is consistent because it has at least one solution. It is also independent because there is only one solution.
Work Step by Step
The equation gives $P$ in terms of $t$, so let's use this expression for $P$ to substitute into the second equation:
$t + 3(\frac{2}{3}t + 5) = 21$
Use distributive property first, paying attention to the signs:
$t + 2t + 15 = 21$
Combine like terms:
$3t + 15 = 21$
Collect constant terms on the right side of the equation:
$3t = 6$
Divide each side of the equation by $3$ to solve for $t$:
$t = 2$
Now that we have the value for $t$, we can substitute it into the first equation to find $P$:
$P = \frac{2}{3}(2) + 5$
Add to solve for $P$:
$P = \frac{4}{3} + 5$
Convert $5$ to a fraction that has $3$ as its denominator so we can add the two fractions:
$P = \frac{4}{3} + \frac{15}{3}$
Add the fractions:
$P = \frac{19}{3}$
The solution to this equation is $t = 2$ and $P = \frac{19}{3}$.
Let's check the answer by substituting both values into one of the original equations to see if both sides are equal:
$2 + 3(\frac{19}{3}) = 21$
Multiply first:
$2 + 19 = 21$
Add:
$21 = 21$
The two sides are equal; therefore, this solution is correct.
This system of equations is consistent because it has at least one solution. It is also independent because there is only one solution.
