Q 329 - Intermediate Algebra

Question

In the following exercises, determine if the following points are solutions to the given system of

equations.

$$\begin{gathered} x + y = 8 \\ y=x-4 \end{gathered}$$

(a) (6,2)

(b) (9.-1)

Step-by-Step Solution

Answer

For (a), the ordered pair,  (6,2)is  a solution of the system of linear equations.

For (b), the ordered pair ,(9,-1) is not a solution of the system of linear equations.

Step 1. Given information

The given linear equations

$$\begin{gathered} x + y = 8 \\ y=x-4 \end{gathered}$$

(a) (6,2)

(b) (9.-1)

Step 2: Checking the values

$$\begin{gathered} \mathrm{First} \mathrm{substitute} x = 6 , \\ y = 2 i\mathrm{nto} \mathrm{the} \mathrm{equation} x + y = 8 \\ x + y = 8 \\ 6+2=8 \\ 8=8 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \\ \mathrm{Also} \mathrm{substitute} x = 6, \\ y = 2 \mathrm{into} \mathrm{the} \mathrm{equatio}n \\ y = x - 4 \\ y = x - 4 \\ 2=6-4 \\ 2=2 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \\ \mathrm{Hnece}, \mathrm{the} \mathrm{ordered} \mathrm{pair} (6,2) \mathrm{is} \mathrm{a} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{linear} \mathrm{equations}. \end{gathered}$$

Step 3: Substitute for another equation

$$\begin{gathered} \mathrm{First} \mathrm{substitut}e x=9, \\ y=-1 \\ \mathrm{into} \mathrm{the} \mathrm{equation} x+y=8 \\ x+y=8 \\ 9-1=8 \\ 8=8 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \\ \mathrm{Also} \mathrm{substitut}e x=9, \\ y=-1 \mathrm{Into} \mathrm{the} \mathrm{equation} \\ y=x-4 \\ y=x-4 \\ -1=9-4 \\ -1=5 \\ \mathrm{This} \mathrm{is} \mathrm{not} \mathrm{true}. \\ \mathrm{Therefore},\mathrm{the} \mathrm{ordered} \mathrm{pair}(9,-1) \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{linear} \mathrm{equations}. \end{gathered}$$