Problem 53 - 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. $$ \left\\{\begin{array}{l} y=-2 x+1 \\ 4 x+2 y=8 \end{array}\right. $$

Step-by-Step Solution

Answer

No solutions; Inconsistent system.

Step 1 - Identify the system of equations

The given system of equations is: 1) \( y = -2x + 1 \) 2) \( 4x + 2y = 8 \)

Step 2 - Rewrite the second equation

Rewrite the second equation to make it easier to compare with the first. \( 4x + 2y = 8 \) Divide every term by 2: \( 2x + y = 4 \)

Step 3 - Compare both equations

The first equation is \( y = -2x + 1 \). Rearrange it to the form \( 2x + y = 1 \). The second equation is \( 2x + y = 4 \).

Step 4 - Analyze the equations

Notice that \( 2x + y = 1 \) and \( 2x + y = 4 \) have the same left-hand side but different right-hand sides. This indicates that the lines are parallel and will never intersect.

Step 5 - Determine the number of solutions

Since the lines are parallel, there are no solutions to the system. Therefore, it is an inconsistent system.

Key Concepts

linear equations

Linear equations are the backbone of algebra and play a central role in many mathematical concepts. These equations form straight lines when graphed on a Cartesian plane. The standard form of a linear equation is given by: \[ ax + by = c \]where \(a\), \(b\), and \(c\) are constants.
Linear equations can help us understand relationships between variables and predict outcomes. For instance, in the problem provided, the given equations are:

• \( y = -2x + 1 \)
• \(4x + 2y = 8\)

By converting and comparing these equations in simpler forms, we can determine their relationships and intersections or lack thereof.

parallel lines

Parallel lines are lines that never meet. They remain the same distance apart over their entire lengths. In the context of linear equations, two lines are parallel if they have the same slope but different y-intercepts.
For the given equations:

• \( y = -2x + 1 \)
• \( 2x + y = 4 \)
After converting and simplifying, we observe they have:
• The same slope: \( -2 \)
• Different y-intercepts: \( 1 \) and \( 4 \)

This means that the lines will not intersect at any point and are parallel.

inconsistent systems

An inconsistent system is one that has no solutions. This happens when the equations representing the system describe lines that do not intersect. In our problem, after simplifying, we get the following equations:

• \( 2x + y = 1 \)
• \( 2x + y = 4 \)

Due to the same slope but different intercepts, these lines are parallel, meaning there's no point that satisfies both equations simultaneously.
Thus, we conclude that the system is inconsistent. Understanding this helps us solve more complex systems and analyze them quickly.