Problem 51 - Elementary Algebra

Question

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations. $$ \left\\{\begin{array}{l} y=\frac{2}{3} x+1 \\ -2 x+3 y=5 \end{array}\right. $$

Step-by-Step Solution

Answer

No solutions; the system is inconsistent.

Step 1: Rewrite the Equations in Standard Form

Make sure both equations are in the form Ax + By = C. The first equation is already in slope-intercept form, so we'll convert it.Equation 1: First equation: \( y = \frac{2}{3} x + 1 \)Multiply both sides by 3 to remove the fraction:\( 3y = 2x + 3 \)Rearrange to standard form Ax + By = C:\( -2x + 3y = 3 \)Equation 2 is already in standard form:\( -2x + 3y = 5 \)

Step 2: Compare the Equations

Compare both equations in standard form:Equation 1: \( -2x + 3y = 3 \)Equation 2: \( -2x + 3y = 5 \)Notice that both equations have the same coefficients for x and y.

Step 3: Determine Consistency and Dependency

Since the equations have the same coefficients but different constants, they represent parallel lines. Parallel lines do not intersect.Thus, the system of equations is inconsistent and has no solutions.

Step 4: Classify the System

Since the system has no solutions, it is classified as 'inconsistent'.

Key Concepts

number of solutions

To determine the number of solutions in a linear system, first look at the equations given. For a system of two linear equations, there are three possible outcomes:
- One solution: If the lines intersect at exactly one point.
- No solution: If the lines are parallel and never intersect.
- Infinite solutions: If the lines are identical and lie on top of each other.

In our exercise, we find that there are no solutions. This happens when the lines are parallel and thus never meet.

standard form

Linear equations can be written in various forms, but the most useful form for comparing equations is the standard form, Ax + By = C. Here, A, B, and C are constants, and x and y are variables.

In the exercise, the first equation was initially in slope-intercept form (y = mx + b). To convert it to standard form, we multiply through to eliminate fractions and rearrange terms to fit the Ax + By = C format:

• Slope-intercept form: \( y = \frac{2}{3} x + 1 \)
• Multiply both sides by 3: \( 3y = 2x + 3 \)
• Rearrange to standard form: \( -2x + 3y = 3 \)

This standard form makes it easier to compare with the second equation given, which is naturally in standard form: \( -2x + 3y = 5 \).

inconsistent systems

A system of equations is called 'inconsistent' if it has no solutions. This means the lines represented by the equations are parallel and will never intersect. In our exercise, after converting both equations to standard form, we see that:
- Equation 1: \( -2x + 3y = 3 \)- Equation 2: \( -2x + 3y = 5 \)

Both equations have the same coefficients for x and y but different constants on the right side. This indicates that the lines are parallel and will never meet, making the system inconsistent. An inconsistent system implies that there are no points (x, y) that satisfy both equations simultaneously.

parallel lines

Parallel lines are lines in a plane that never intersect. They have the same slope but different y-intercepts. In the context of linear systems, parallel lines signify that the system of equations has no common solution. In our exercise, we determined that both lines have the same slope by comparing their standard forms:

• Equation 1: \( -2x + 3y = 3 \)
• Equation 2: \( -2x + 3y = 5 \)

The same coefficients for x and y (\( -2 \) and \( 3 \) respectively) but different constants (3 and 5) confirm that these lines are parallel. Therefore, they will never intersect, representing a real-world scenario where two paths run parallel indefinitely without crossing. This visual understanding aids in grasping why no solutions exist for these lines.