Q 230 - Intermediate Algebra

Question

Solve the system of equations by graphing and by substitution. Which method do you prefer? Why? 

$$\begin{gathered} x+y=10 \\ x-y=6 \end{gathered}$$

Step-by-Step Solution

Answer

The solution of the linear equations are:

(8,2)

Step 1. Given information

The given linear equations are :$$\begin{gathered} x+y=10 \\ x-y=6 \end{gathered}$$

 

Step 2. Graphing method

First, solve both of these equations for y such that their slopes and y intercepts may be easily graphed.

And find the slope and y-intercept by solving the first equation for y.

$$\begin{gathered} x + y = 10 \\ y = - x + 10 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} m = - 1 \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} b = 10 \end{gathered}$$

Again find the slope and y-intercept by solving the Second equation for y.  

$$\begin{gathered} x - y = 6 \\ - y = - x + 6 \\ y = x - 6 \\ \mathrm{Here} \mathrm{the} \mathrm{slope} \mathrm{is} m = 1 \\ \mathrm{And} \mathrm{the} \mathrm{y}-\mathrm{intercept} \mathrm{is} b = - 6 \end{gathered}$$

The Graph for both lines

$$\begin{gathered} \mathbf{Check}\mathbf{:}\mathbf{ } \\ \mathrm{First} \mathrm{substitute} x=8, y=2 \mathrm{Into} \mathrm{the} \mathrm{equation} x+y=10 \\ x+y=10 \\ 8+2=10 \\ 10-10 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \\ \mathrm{Also} \mathrm{substitute} x=8,2 \mathrm{into} \mathrm{the} \mathrm{equation} x-y=6 \\ x-y=6 \\ 8-2=6 \\ 6=6 \\ \mathrm{This} \mathrm{is} \mathrm{true}. \\ \mathrm{Therefore}, \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{linear} \mathrm{equations} \mathrm{is} (8,2) \end{gathered}$$

Step 3. Substitution method

$$\begin{gathered} \mathrm{First} \mathrm{solve} \mathrm{one} \mathrm{equation} \mathrm{for} \mathrm{one} \mathrm{variable}. \\ \mathrm{Solve} \mathrm{the} \mathrm{first} \mathrm{equation} \mathrm{for} \mathrm{y}. \\ x + y = 10 \\ y = 10 - x \\ \mathrm{Substitute} y = 10 - x i\mathrm{n} \mathrm{second} \mathrm{equation} \\ x + y = 10 \mathrm{and} \mathrm{solve} \mathrm{for} x. \\ x - y = 6 \\ x - 10 + x = 6 \\ 2x - 10 = 6 \\ 2x = 6 + 10 \\ 2x = 16 \\ x = 16/2 \\ =8 \end{gathered}$$

$$\begin{gathered} \mathrm{Substitute} x = 8 \\ \mathrm{into} x + y = 10 \mathrm{and} \mathrm{solve} \mathrm{for} \mathrm{y}. \\ x + y = 10 \\ 8 + y = 10 \\ y = 10 - 8 \\ y = 2 \end{gathered}$$