Question

Determine whether an ordered pair is a solution of a system of equations. In the following exercises, determine if the following points are solutions to the given system of equations.

$\left\{\begin{array}{l}x+y=1\\ y=\frac{2}{5}x\end{array}\right.$

(a)$\left(\frac{5}{7},\frac{2}{7}\right)$

(b)$\left(5,2\right)$

Short answer

Part (a) An ordered pair $\left(\frac{5}{7},\frac{2}{7}\right)$is a solution.

Part (b) An ordered pair width="34">(5,2)is not a solution.

Step-by-step solution

Part (a) Step 1. Given information

Consider the system of equations.

$\left\{\begin{array}{l}x+y=1\\ y=\frac{2}{5}x\end{array}\right.$

Part (a) Step 2. Determine whether an ordered pair 57,27 is a solution to the given system of two equations.

Substitute the values of the variables into each equation to determine whether an ordered pair is a solution to a given system of two equations. It is a solution to the system if the ordered pair makes both equations true.

Substitute $\frac{5}{7}$for $x$and $\frac{2}{7}$for localid="1647515260801" $y$into $x+y=1$.

$\begin{array}{rcl}\frac{5}{7}+\frac{2}{7}& =& 1\\ \frac{5+2}{7}& =& 1\\ \frac{7}{7}& =& 1\\ 1& =& 1\left(\text{True}\right)\end{array}$

Substitute $\frac{5}{7}$for $x$and $\frac{2}{7}$for $y$into $y=\frac{2}{5}x$.

$\begin{array}{rcl}\frac{2}{7}& =& \frac{2}{5}·\frac{5}{7}\\ \frac{2}{7}& =& \frac{2}{7}\left(\text{True}\right)\end{array}$

Conclude that the ordered pair $\left(\frac{5}{7},\frac{2}{7}\right)$made both equations true.

Thus, $\left(\frac{5}{7},\frac{2}{7}\right)$is a solution to the given system.

Part (b) Step 1. Determine whether an ordered pair (5,2) is a solution to the given system of two equations.

Substitute $5$for $x$and $2$for $y$into $x+y=1$.

localid="1647515527913" $\begin{array}{rcl}5+2& =& 1\\ 7& =& 1\left(\text{False}\right)\end{array}$

Substitute $5$for $x$and $2$for $y$into $y=\frac{2}{5}x$.

$\begin{array}{rcl}2& =& \frac{2}{5}\left(5\right)\\ 2& =& 2\left(\text{True}\right)\end{array}$

Conclude that the ordered pair $(5,2)$made one equation true and other equation false.

Thus, localid="1647515621543" $(5,2)$is not a solution to the given system.

Hence, $\left(\frac{5}{7},\frac{2}{7}\right)$is a solution and $(5,2)$is not a solution to the given system of equations.