Problem 1 - Intermediate Algebra

Question

For Problems \(1-16\), use the graphing approach to determine whether the system is consistent, the system is inconsistent, or the equations are dependent. If the system is consistent, find the solution set from the graph and check it. (Objective 1) $$ \left(\begin{array}{r} x-y=1 \\ 2 x+y=8 \end{array}\right) $$

Step-by-Step Solution

Answer

The system is consistent, and the solution is (3, 2).

Step 1: Understand the Graphing Approach

To determine whether the system is consistent, inconsistent, or the equations are dependent, graph the equations and observe the intersection point (if any). Each line represents an equation in the system.

Step 2: Graph the First Equation

The first equation is \( x - y = 1 \). Rearrange it to the slope-intercept form \( y = mx + b \): \( y = x - 1 \). This equation represents a line with a slope of 1 and a y-intercept of -1. Plot this line on the graph.

Step 3: Graph the Second Equation

The second equation is \( 2x + y = 8 \). Rearrange it to \( y = -2x + 8 \). This line has a slope of -2 and a y-intercept of 8. Plot this line on the same graph as the first equation.

Step 4: Analyze Intersection Point

Observe where the two lines intersect on the graph. If they intersect at a single point, the system is consistent and the coordinates of this point are the solution to the system. If they never intersect (i.e., they are parallel), the system is inconsistent. If they are the same line (i.e., coincide), the equations are dependent.

Step 5: Identify Consistency and Find Solution Set

On the graph, the lines \( y = x - 1 \) and \( y = -2x + 8 \) intersect at the point (3, 2). Therefore, the system is consistent, and the solution set is \( (3, 2) \).

Step 6: Verify the Solution

To ensure the solution is correct, substitute \( x = 3 \) and \( y = 2 \) back into both original equations. The first equation becomes \( 3 - 2 = 1 \), which is true. The second equation becomes \( 2(3) + 2 = 8 \), which is also true, confirming that (3, 2) is indeed the correct solution.

Key Concepts

Graphing Method in Algebra

In algebra, the graphing method is a way to visually solve a system of linear equations. By graphing, we depict each equation of the system as a line on a coordinate plane. This helps us to easily visualize where the solutions might be. For this approach, each equation is transformed into slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

To use the graphing method:

• Convert each equation to the slope-intercept form if they aren't already.
• Draw each line on the same graph by plotting the y-intercept, then using the slope to find another point on the line.
• Look for the intersection point, which represents the solutions.

If the lines meet at exactly one point, this point is the solution to the system. Graphing is particularly useful for visual learners and helps to grasp the concept of solving equations by providing a clear picture of their solutions.

Solution of Equations

Finding the solution of equations in a system means identifying the values of the variables involved that satisfy all equations simultaneously. In the context of the graphing method, the solution is the point where the lines intersect. This intersection point is where each equation's graph "agrees" or "satisfies" the relationship expressed by the system, meaning it holds true for all equations.

Let's consider the example given:

• The first equation \( x - y = 1 \) becomes \( y = x - 1 \).
• The second equation \( 2x + y = 8 \) becomes \( y = -2x + 8 \).
• Plotting these reveals an intersection at point \( (3, 2) \).

The coordinates of this point, \( (3, 2) \), are the solution to the system. This means if you substitute \( x = 3 \) and \( y = 2 \) back into the original equations, both will hold true, confirming the correct solution.

Consistent and Inconsistent Systems

In systems of linear equations, the terms consistent and inconsistent describe the nature of solutions:

• Consistent system: This system has at least one solution. Graphically, this means that the lines intersect at a specific point, or they are the same line (coinciding over all points).
• Inconsistent system: This indicates the system has no solutions. Graphically, the lines are parallel and will never meet since no point satisfies all equations simultaneously.
• Dependent equations: The equations describe the same line, resulting in infinite solutions because every point along the line is a solution.

In the exercise, after plotting the two lines \( y = x - 1 \) and \( y = -2x + 8 \), they intersect at the point \( (3, 2) \). Therefore, this system is categorized as consistent, with a single solution at the point of intersection. By understanding whether a system is consistent or inconsistent, we can determine if solutions exist and their nature, which is vital in problem-solving across various algebraic contexts.