In Exercises 19–30, solve each system by the addition method. x + y = 1 x - y = 3

Verified step by step guidance

1. Write down the system of equations clearly: $x+y=1x-y=3$.
2. Add the two equations together to eliminate $y$: $(x\; +\; y)\; +\; (x\; -\; y)\; =\; 1\; +\; 3$. Notice that $+y$ and $-y$ cancel out.
3. Simplify the resulting equation: $2x\; =\; 4$. This gives you an equation with only one variable.
4. Solve for $x$ by dividing both sides of the equation by 2: $x\; =\; \backslash (\backslash frac\{4\}\{2\}\backslash )$.
5. Substitute the value of $x$ back into one of the original equations (for example, $x\; +\; y\; =\; 1$) to solve for $y$. Rearrange to find $y\; =\; 1\; -\; x$.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. In this problem, the system has two equations with two variables, x and y.

Addition Method (Elimination Method)

The addition method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable. By aligning terms and combining equations, one variable cancels out, simplifying the system to a single-variable equation.

Solving for Variables

After eliminating one variable using the addition method, solve the resulting single-variable equation. Substitute this solution back into one of the original equations to find the value of the other variable, ensuring both equations are satisfied.