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In algebra, a system of equations is a collection of two or more equations with the same set of variables. Here, the given system of equations includes:
\( x + y = 3 \) and \( y = 4 - x \). The goal is to find the values of the variables that satisfy all equations simultaneously. In other words, we are looking for a point (a specific x and y) where both equations intersect. If such a point does not exist, then the system has no solution. It is crucial to understand that the equations represent lines in a coordinate plane, and we aim to find their intersection. Sometimes, the equations might [[not have any common solutions at all]](https://www.splashmath.com/math-vocabulary/algebra), leading to parallel lines that do not meet, indicating inconsistent systems.

The substitution method is a technique to solve systems of equations. Here’s how we use it:
First, solve one of the equations for one variable. In our example, the second equation is already solved for y: \( y = 4 - x \).
Next, substitute this expression for y into the first equation: \( x + (4 - x) = 3 \). This substitution simplifies the system to a single equation in one variable.
We then simplify and solve for x: \( x + 4 - x = 3 \), resulting in a simplified form: \( 4 = 3 \).
Notice that this form is a contradiction, indicating no solution. This contradiction means these lines are parallel and do not intersect.

Graphing calculators are valuable tools to visualize and verify solutions to systems of equations. You can input both equations into the graphing calculator:
\begin{aligned} \( x + y = 3 \) \( y = 4 - x \) \ \text{After entering the equations, look at the graph. Both equations will appear as lines on the coordinate plane, and you can see if they intersect.}
Since we know from algebraic manipulation \( 4 = 3 \) is false, confirming the absence of solutions, the graph will show parallel lines that never intersect. Thus, the lack of an intersection point on the graph supports the conclusion that there are no solutions to the system. Always use graphing calculators to double-check your algebraic work.