Question

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

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

Short answer

The solution of the linear equations are:

(8,2)

Step-by-step solution

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

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

role="math" localid="1644425289016" $$\begin{gathered} \mathrm{Substitute}x=8 \\ \mathrm{into}x+y=10\mathrm{and}\mathrm{solve}\mathrm{for}\mathrm{y}. \\ x+y=10 \\ 8+y=10 \\ y=10-8 \\ y=2 \end{gathered}$$