Question

writing solve the system of equations using subtraction. use pencil and paper. describe two different ways to solve the system.$x+y=3$$x-y=1$the solution is $\boldsymbol{\square}$. (type an ordered pair.)

Explanation:

Step1: Add the two equations

$$(x + y) + (x - y) = 3 + 1$$
$$2x = 4$$

Step2: Solve for $x$

$$x = \frac{4}{2} = 2$$

Step3: Substitute $x=2$ into first equation

$$2 + y = 3$$

Step4: Solve for $y$

$$y = 3 - 2 = 1$$

Alternative Method 1 (Subtraction):

Step1: Subtract second from first equation

$$(x + y) - (x - y) = 3 - 1$$
$$2y = 2$$

Step2: Solve for $y$

$$y = \frac{2}{2} = 1$$

Step3: Substitute $y=1$ into first equation

$$x + 1 = 3$$

Step4: Solve for $x$

$$x = 3 - 1 = 2$$

Alternative Method 2 (Isolate then Substitute):

Step1: Isolate $x$ from second equation

$$x = y + 1$$

Step2: Substitute into first equation

$$(y + 1) + y = 3$$
$$2y + 1 = 3$$

Step3: Solve for $y$

$$2y = 2 \implies y = 1$$

Step4: Substitute $y=1$ to find $x$

$$x = 1 + 1 = 2$$

Answer:

$(2, 1)$

Two ways to solve the system:

1. Elimination by Addition: Add the two equations to eliminate $y$, solve for $x$, then substitute back to find $y$.
2. Substitution Method: Isolate a variable (e.g., $x$) from one equation, substitute it into the other equation, and solve for the remaining variable.