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.

$\left\{\begin{array}{l}7x-4y=-1\\ -3x-2y=1\end{array}\right.$

(a)$(1,2)$

(b)$(1,-2)$

Short answer

Part (a) An ordered pair $(1,2)$is a solution.

Part (b) An ordered pair $(1,-2)$is not a solution.

Step-by-step solution

Part (a) Step 1. Given information

Consider the system of equations.

$\left\{\begin{array}{l}7x-4y=-1\\ -3x-2y=1\end{array}\right.$

Part (a) Step 2. Determine whether an ordered pair (1,2) 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 $1$for $x$and $2$for $y$into $\begin{array}{rcl}7x-4y& =& -1\end{array}$.

localid="1647494048951" $\begin{array}{rcl}7\left(1\right)-4\left(2\right)& =& -1\\ 7-8& =& -1\\ -1& =& -1(\text{True)}\end{array}$

Substitute $1$for $x$and $2$for $y$into $-3x-2y=1$.

$\begin{array}{rcl}-3\left(1\right)-2\left(2\right)& =& 1\\ -3-4& =& 1\\ -7& =& 1\left(\text{False}\right)\end{array}$

Conclude that the ordered pair $(1,2)$made one equation true and other equation false.

Thus, $(1,2)$is not a solution to the given system.

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

Substitute $1$for $x$and $-2$for $y$into $\begin{array}{rcl}7x-4y& =& -1\end{array}$.

localid="1647494079736" $\begin{array}{rcl}7\left(1\right)-4(-2)& =& -1\\ 7+8& =& -1\\ 15& =& -1(\text{False)}\end{array}$

Substitute $1$for $x$and $-2$for $y$into $-3x-2y=1$.

$\begin{array}{rcl}-3\left(1\right)-2(-2)& =& 1\\ -3+4& =& 1\\ 1& =& 1\left(\text{True}\right)\end{array}$

Conclude that the ordered pair $(1,-2)$made one equation true and other equation false.

Thus, $(1,-2)$is not a solution to the given system.

Hence, $(1,2)$and $(1,-2)$ are not the solutions to the given system of equations.