Match the system of equations with one of the graphs ( a ) -...

Match the system of equations with one of the graphs ( $a$ ) - $(f),$ which follow.
A.(GRAPH CANNOT COPY)
B.(GRAPH CANNOT COPY)
C.(GRAPH CANNOT COPY)
D.(GRAPH CANNOT COPY)
E.(GRAPH CANNOT COPY)
F.(GRAPH CANNOT COPY)
$$\begin{aligned}
&4 x-2 y=5\\
&6 x-3 y=-10
\end{aligned}$$

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Key Concepts

System of Linear Equations

A system of linear equations is a set of two or more equations with the same variables. The goal is to find values for the variables that satisfy every equation in the system simultaneously, which can be visualized by identifying the points where the lines representing each equation intersect on a graph.

Slope-Intercept Form

The slope-intercept form of a linear equation, given as y = mx + b, is a useful way to quickly determine the slope and y-intercept of a line. Recognizing the slope in this form is central to understanding the behavior of a line and how it relates to other lines in a system.

Parallel Lines

Parallel lines are lines in the plane that have identical slopes but different y-intercepts. Because they maintain the same constant vertical separation, they will never cross, which is an important characteristic when analyzing systems of equations graphically.

Inconsistent Systems

An inconsistent system is one in which the equations do not share a common solution; in a graphical context, this corresponds to lines that do not intersect. Identifying an inconsistent system is critical for understanding that no single set of variable values can satisfy all equations simultaneously.

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