Question

solve the system of equations. if the system has no solution, say that it is inconsistent. graph the lines of the system.\

$$\begin{cases}x + y = 3\\\\x - y = 1\\end{cases}$$

\dots select the correct choice below and, if necessary, fill in any answer boxes within your choice.\bigcirc a. the solution is x = \square and y = \square. (type integers or simplified fractions.)\bigcirc b. there are infinitely many solutions. using ordered pairs, they can be expressed as \\{(x,y)| x = \square, y is any real number\\}. (simplify your answer. type an expression using y as the variable as needed.)\bigcirc c. the system is inconsistent.

Explanation:

Step1: Add the two equations

We have the system:

$$\begin{cases} x + y = 3 \\ x - y = 1 \end{cases}$$

Adding the left - hand sides and the right - hand sides of the two equations: $(x + y)+(x - y)=3 + 1$.
Simplifying the left - hand side: $x + y+x - y = 2x$, and the right - hand side is $4$. So we get the equation $2x=4$.

Step2: Solve for x

Divide both sides of the equation $2x = 4$ by $2$: $x=\frac{4}{2}=2$.

Step3: Substitute x = 2 into one of the original equations

Substitute $x = 2$ into the first equation $x + y=3$. We get $2 + y=3$.
Subtract $2$ from both sides of the equation: $y=3 - 2=1$.

Answer:

A. The solution is $x = 2$ and $y = 1$.