Question

4p - 2t = 6
8p + 6t = 12 + 2t
which of the following accurately describes all solutions to the system of equations shown?
choose 1 answer:
(a) p = \frac{3}{2} and t = 0
(b) p = 3 and t = 3
(c) there are infinite solutions to the system.
(d) there are no solutions to the system.

Explanation:

Step1: Simplify the second - equation

First, simplify the equation \(8p + 6t=12 + 2t\). Subtract \(2t\) from both sides: \(8p+6t - 2t=12+2t - 2t\), which gives \(8p + 4t=12\). Then divide the entire equation by \(4\): \(\frac{8p + 4t}{4}=\frac{12}{4}\), resulting in \(2p + t = 3\), or \(t=3 - 2p\).

Step2: Substitute into the first equation

Substitute \(t = 3 - 2p\) into the first equation \(4p-2t = 6\). We get \(4p-2(3 - 2p)=6\). Expand the left - hand side: \(4p-6 + 4p=6\). Combine like terms: \(8p-6 = 6\). Add \(6\) to both sides: \(8p-6 + 6=6 + 6\), so \(8p=12\). Divide both sides by \(8\): \(p=\frac{12}{8}=\frac{3}{2}\).

Step3: Find the value of \(t\)

Substitute \(p = \frac{3}{2}\) into \(t = 3 - 2p\). Then \(t=3-2\times\frac{3}{2}=3 - 3=0\).

Answer:

A. \(p=\frac{3}{2}\) and \(t = 0\)