91Ó°ÊÓ

Visualizing equations as graphs can help us understand how lines relate to each other on the coordinate plane. Every linear equation can be graphed as a straight line. By examining these graphs, we can determine whether two lines are parallel, intersecting, or coinciding.
In the given exercise, we have two linear equations:

• Equation 1: \(4y + 5x = 1\)
• Equation 2: \(10x + 2y = 2\)

Once the equations are expressed correctly, we can graph them to see if they intersect, are parallel, or coincide.
Drawing these lines involves plotting points or using the slope and intercept directly. Observing these interactions can clarify the relationships between the equations, providing a visual insight that is sometimes clearer than numbers alone.
In this situation, the lines will cross each other, as confirmed by their differing slopes, which we found by analyzing the equations further.

The slope-intercept form of a linear equation is a very useful way to express line equations. This form is \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept.Converting equations into this format makes it easy to identify these two crucial components of any line:

• The slope \(m\) shows how steep the line is. It tells us how much \(y\) changes for a unit change in \(x\). A positive slope means the line rises, while a negative slope means it falls.
• The y-intercept \(b\) is where the line crosses the y-axis. It gives a starting point for drawing the line on the graph.

In our problem, rearranging the equations

• from \(4y + 5x = 1\) to \(y = -1.25x + 0.25\), and
• from \(10x + 2y = 2\) to \(y = -5x + 1\)

helps us see that they have different slopes and y-intercepts. This implies distinct lines that are not parallel.

A system of linear equations involves two or more equations that you consider simultaneously. The goal is often to find a common solution or point that satisfies all equations in the system.
In graphical terms, this means finding the point where lines representing these equations intersect on a plane. This intersection point provides the values of \(x\) and \(y\) that work for both equations.
For the given equations:

• First equation: \(y = -1.25x + 0.25\)
• Second equation: \(y = -5x + 1\)

We determine that they are not parallel. Thus, there must be a unique intersection point.
Systems of equations can have one intersection (one solution), no intersections (parallel lines - meaning no solution), or infinitely many intersections (coinciding lines). Here, because the lines are distinct and cross at one point, the system has exactly one solution.