Q 328 - Intermediate Algebra

Question

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

$$\begin{gathered} \left\{\begin{array}{l} x+3y=-9 \\ 2x - 4y = 12\\ \end{array}\right. \end{gathered}$$

(a) (-3,-2)

(b)(0, -3)

Step-by-Step Solution

Answer

For (a), the ordered pair, (-3,-2) is not a solution of the system of linear equations.

For (b), the ordered pair ,(0,-3) is  a solution of the system of linear equations.

Step 1: Given information

Given the system of equations 

$$\begin{gathered} \left\{\begin{array}{l} x+3y=-9 \\ 2x - 4y = 12\\ \end{array}\right. \end{gathered}$$

Ans the points are

(a) (-3,-2)

(b)(0,-3)

Step 2. Checking for values

$$\begin{gathered} \mathrm{First} \mathrm{substitute} x=-3, \\ y = - 2 \mathrm{into} \mathrm{the} \mathrm{equation} \\ x + 3y = - 9 \\ x + 3y = - 9 \\ -3+3(-2)=-9 \\ -3-6=-9 \\ -9=-9 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \\ \mathrm{Also} \mathrm{substitute} x = - 3, \\ y = 2 \mathrm{into} \mathrm{the} \mathrm{equation} \\ 2x - 4y = 12 \\ 2x - 4y = 12 \\ 2(- 3) - 4(- 2) = 12 \\ -6+8=12 \\ 2=12 \\ \mathrm{This} \mathrm{is} \mathrm{not} \mathrm{true}. \\ \mathrm{Hence} , \mathrm{the} \mathrm{ordered} \mathrm{pair} (-3,-2) \mathrm{is} \mathrm{not} \mathrm{a} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{linear} \mathrm{equations}. \end{gathered}$$

Step 3: Substitute for another equation

$$\begin{gathered} \mathrm{First} \mathrm{substitute} x=0, \\ y=-3 \mathrm{Into} \mathrm{the} \mathrm{equation} x+3y=-9 \\ x+3y=-9 \\ 0+3(-3)=-9 \\ 0-9=-9 \\ -9=-9 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \\ \mathrm{Also} \mathrm{substitute} x=0,y=-3 \mathrm{Into} \mathrm{the} \mathrm{equati}on 2x-4y=12 \\ 2x-4y=12 \\ 2\left(0\right)-4(-3)=12 \\ 0+12=12 \\ 12=12 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \\ \mathrm{Therefore},\mathrm{the} \mathrm{ordered} \mathrm{pair}(0,-3)\mathrm{is} \mathrm{a} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{linear} \mathrm{equations}. \end{gathered}$$