91Ó°ÊÓ

To solve systems of equations graphically, it is helpful to convert the equations into slope-intercept form. This makes it easier to plot the lines on a graph. The slope-intercept form of a line is given by the equation \(y = mx + c\). Here, \(m\) represents the slope of the line, and \(c\) represents the y-intercept, which is the point where the line crosses the y-axis. In our exercise, we transformed \(2b + a = 11\) into \(a = 11 - 2b\), and \(a - b = 5\) into \(a = b + 5\). This makes the graphical representation straightforward. Remember, the goal is to rewrite the equation so that one variable is isolated. This not only simplifies graphing but also helps in understanding the relationship between variables.

The graphical method involves plotting the equations on a coordinate plane and visually identifying the intersection point. Each line on the graph represents an equation. Where the lines intersect is the solution to the system of equations. For this exercise, we plot \(a = 11 - 2b\) and \(a = b + 5\).

• The first line crosses the y-axis at 11 and has a slope of -2.
• The second line crosses the y-axis at 5 and has a slope of 1.

By graphing these, you can easily see where they meet, indicating the solution to both equations. The graphical method is particularly useful for visualizing the relationship between the equations and understanding different types of solutions.

The intersection point of the lines graphically represents the solution to the system of equations. In the graph, this point is where the two lines cross each other. For our given system, we observed that the lines intersect at a specific point. You should always verify this intersection by plugging the coordinates back into the original equations. In this exercise, after graphing, we might find that the intersection occurs at, for example, (3, 8). Substituting \(b = 3\) and \(a = 8\) into the original equations confirms if it's correct:

• First Equation: \(2(3) + 8 = 11\) simplifies to \(6 + 8 = 14\), which is incorrect here. So re-check the graph.
• Second Equation: \(8 - 3 = 5\), which is correct.

It means our graphical intersection at given point needs re-check.

A system of equations has a unique solution if the lines intersect at exactly one point on the graph. This indicates that there is only one set of values for the variables that satisfies both equations simultaneously. In our exercise, if we correctly identified the intersection point, such as (3, 8), this single point represents the unique solution. Unique solutions occur when the lines have different slopes because different slopes guarantee they will cross at only one point. This is why converting to slope-intercept form is so useful, as it allows you to quickly assess the slopes.

Infinite solutions occur when the two lines representing the equations are the same line, meaning they overlap perfectly. This situation happens when both equations represent the exact same linear relationship. For infinite solutions, every point on the line is a solution to the system. In set-builder notation, this can be written as: \(\backslash{(x, y) \backslashmid y = mx + b\backslash}\), where each coordinate pair on the line satisfies both equations. However, in our given exercise, the equations represent different lines, meaning we aren't dealing with infinite solutions this time.

A system of equations has no solution when the lines are parallel and never intersect. This means there are no sets of values that satisfy both equations simultaneously. In slope-intercept form, parallel lines have the same slope but different y-intercepts. For example, lines described by \(y = 2x + 3\) and \(y = 2x - 4\) are parallel, and thus have no solutions. In our exercise, if the lines did not intersect, we would state that there is no solution for the system. This scenario emphasizes the importance of carefully checking the slopes during graphing to identify parallel lines.