In Exercises 43-50, find the slope and the y -intercept for...

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions. $$\left\{\begin{array}{l} y=\frac{1}{2} x-3 \\ y=\frac{1}{2} x-5 \end{array}\right.$$

In Exercises $43-50$, find the slope and the $y$ -intercept for the graph of each equation in the given system. Use this information (and not the equations' graphs) to determine if the system has no solution, one solution, or an infinite number of solutions.
$$\left\{\begin{array}{l}
y=\frac{1}{2} x-3 \\
y=\frac{1}{2} x-5
\end{array}\right.$$

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Key Concepts

Slope-Intercept Form

This is a way to write the equation of a line in the form y = mx + b where m represents the slope and b represents the y-intercept. It provides a direct method to read off the steepness of the line and its crossing point with the y-axis, which is crucial for graphical analysis and comparisons.

Slope

The slope, indicated by m in the slope-intercept form, measures the rate of change of the line. It shows how much the value of y increases or decreases for each unit increase in x, thereby determining the line's direction and steepness.

Y-intercept

The y-intercept, represented by b in the equation y = mx + b, is the point where the line crosses the y-axis (x = 0). It is an important feature for graphing a line and understanding its vertical positioning within the coordinate plane.

Parallel Lines

Lines that have equal slopes but different y-intercepts are parallel. This means that they have the same direction and steepness, but they are vertically offset from each other, which implies that they will never intersect on the plane.

Inconsistent Linear Systems

A system of linear equations is inconsistent when the equations represent parallel lines, thereby having no point of intersection. This means that there is no set of (x, y) values that satisfies both equations simultaneously, resulting in no solution for the system.

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