Question

Use substitution to solve each system of equations.

$\begin{array}{l}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{9}\\ \mathbf{x}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{1}\end{array}$

Short answer

The solution of the system of equations is $\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{)}$.

Step-by-step solution

Step 1. State the concept of substitution method to solve system of equations.

The substitution method is used to find the set of solutions for the given system of equations. It is the method in which one variable is first solved in the form of another variable.

So, the value of one variable in the form second variable is put on the second equation. It will give the solution of the one variable. And then solve it for another variable.

Step 2. Solve at least one equation for one variable and substitute.

The given system of equation is:

$x+3y=9$ … (1)

$x+y=1$ … (2)

Note that in equation (2), variable $y$can be easily converted in terms of $x$. $\begin{array}{c}x+y=1\\ x+y-x=1-x\\ y=1-x\end{array}$

Now put localid="1647508520134" $y=1-x$$y$in eq. (1):

$\begin{array}{l}x+3y=9\\ x+3(1-x)=9\text{Substitute}y=1-x\\ x+3-3x-3=9-3\text{Combineliketerms}\\ -2x=6\\ x=-3\text{Divideeachsideby}-2\end{array}$

Step 3. Solve for y.

Substitute $-3$for $x$into the equation (2) and solve for $y$.

$\begin{array}{c}x+y=1\\ (-3)+y=1\\ y=1+3\\ =4\end{array}$

The solution is $(-3,4)$.