91Ó°ÊÓ

In algebra, the slope-intercept form is a way to describe the equation of a straight line. It follows the format of \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) denotes the y-intercept, the point where the line crosses the y-axis.
The slope \(m\) shows how steep the line is. A positive slope means the line goes up, while a negative slope means it goes down.
For example, in the first equation of our system, which is originally \(x - 3y = 5\), we can rearrange it to slope-intercept form:

\[-3y = -x + 5\]
• Divide everything by -3:
\[y = \frac{1}{3}x - \frac{5}{3}\]
• So, the slope \(m = \frac{1}{3}\), and the y-intercept \(b = -\frac{5}{3}\).

For the second equation \(2x + y = 8\), it's already quite close to the slope-intercept form. Subtract \(2x\) from both sides:
\[y = -2x + 8\]

Here, the slope \(m = -2\), and the y-intercept \(b = 8\).

Notice how translating to slope-intercept form makes it easier to compare the slopes and y-intercepts of the lines.

Intersecting lines are lines that cross each other at a single point on a graph. This happens when two lines have different slopes.
In our case, the first equation from the slope-intercept form has a slope of \(\frac{1}{3}\), while the second has slope \(-2\). Since these slopes are different, the lines will intersect.

The point where they intersect represents the single solution to the system of equations.
The pair of intersecting lines indicates that:
• The system has one solution.
• The equations in the system are independent.
• The system is consistent.
The visualization could be thought of as a 'X' shape on a graph, where each line crosses at a specific point.

A system of equations consists of two or more equations that share the same set of variables. The main goal is to find the values of these variables that satisfy all the equations simultaneously.
In our example, we have two linear equations:

• \(x - 3y = 5\)
• \(2x + y = 8\)

To solve such a system, you can use various methods, including:

• Graphing
• Substitution
• Elimination

In this case, converting to slope-intercept form helps us graph and visualize the equations accurately. The solution to the system is the point where both lines intersect.
Understanding the nature of the slopes can determine how many solutions exist without graphing it: If the slopes are the same (and y-intercepts different), lines are parallel with no solution. If slopes are different, lines intersect, having one solution.

When a system of linear equations has one solution, it means there is exactly one point where both lines intersect. This point satisfies both equations at the same time.
In our example, the lines represented by:

• \[y = \frac{1}{3}x - \frac{5}{3}\]
• \[y = -2x + 8\]

Intersect at exactly one point because they have different slopes. This unique intersection point is the solution to the system. It can be found algebraically by setting the equations equal to each other and solving for x and y.
For instance:

\[\frac{1}{3}x - \frac{5}{3} = -2x + 8\]

Solve for \(x\) and substitute back to find \(y\). This unique point is the system's one solution.

A consistent system of equations is one that has at least one solution. In our case, because the two lines intersect at one point, this system is consistent.

An inconsistent system, by contrast, has no points of intersection and thus no solutions.
Different scenarios for consistent systems include:
• Parallel lines with the same slope and different y-intercepts, showing no solution.
• Identical lines (infinitely many solutions).
• Intersecting lines (one solution).
In our example, we witnessed a consistent system through one solution, meaning the system's equations are fulfilled at exactly one single point.

Independent equations in a system mean that each equation describes a unique line. These lines intersect exactly at one point, which is the case for our system.
One point of intersection means the system has one unique solution. By contrast, dependent equations describe the same line and have infinitely many solutions.

For our example:
• The equations \(y = \frac{1}{3}x - \frac{5}{3}\) and \(y = -2x + 8\) intersect at a single point.
This indicates two independent equations leading to one solution. Independent equations ensure no overlap beyond their point of intersection, further validating the system is both consistent and independent.