91Ó°ÊÓ

An inconsistent system of equations occurs when there are no solutions because the lines representing the system are parallel and never intersect. Imagine two lines on a graph that never touch each other. These lines will have the same slope but different y-intercepts.
For example, consider the equations:
$$\begin{array}{r} 3x + 2y = 7 \ 3x + 2y = 5 \ \text{} \ \text{} \ \text{} \ \text{} \ \text{} \ \text{} \ \text{} \ \text{} No matter what values you choose for x and y, you can't satisfy both equations alongside one another. If you were to graph these, they would be two parallel lines.
Always look for differing constants after the same linear combination of variables. If you find these, then the system is inconsistent.

Systems with infinitely many solutions occur when the equations describe the same line. This means every point on one line is also a point on the other.
For instance, consider the system:
$$\begin{array}{r} 7x + 2y = 6 \ 14x + 4y = 12 \ \text{} \ \text{} \ \text{} \ \text{} \ \text{} \ \text{} \ \text{} \ \text{} $$
By simplifying the second equation, you get:
$$7x + 2y = 6$$
These two equations are the same. This means every point that satisfies the first equation also satisfies the second equation.
When solving, we set y as arbitrary and solve for x. So for the given system:

• Express y as any real number.
• Then solve for x using y.

In this case:
$$ x = \frac{6 - 2y}{7}$$
Therefore, our solution set includes all the points \( (x, y) \) where x and y satisfy this equation.

Linear equations in two variables are of the form \( ax + by = c \). These graphs represent straight lines on the coordinate plane.
The system of linear equations involves finding the set of values \( x \) and \( y \) that satisfy all equations in the system simultaneously.
They can have:

• One solution (lines intersect at one point).
• Infinitely many solutions (lines are the same).
• No solutions (lines are parallel).

When solving these systems, it's helpful to:

• Graph the equations to visualize their relationship.
• Use substitution or combination methods to find solutions.

For instance, in our exercise:
$$\begin{array}{r} 7x + 2y = 6 \ 14x + 4y = 12 \ \text{} \ \text{} \ \text{} \ \text{} $$
We found they describe the same line, indicating infinitely many solutions. Simplifying, substituting, or graphing can help you identify these relationships clearly.