Problem 57 - 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. $$ \left\\{\begin{array}{l} 4 x+2 y=10 \\ 4 x-2 y=-6 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has exactly one solution and is classified as consistent and independent.

Step 1: Write the System of Equations

The system of equations is given by: \[\begin{cases} 4x + 2y = 10 \ 4x - 2y = -6 \end{cases}\]

Step 2: Simplify the System

To simplify, observe the coefficients of the variables. We can add the two equations to eliminate one of the variables.

Step 3: Add the Equations

Add the two equations together: \[(4x + 2y) + (4x - 2y) = 10 + (-6)\] This simplifies to: \[8x = 4\]

Step 4: Solve for x

Divide both sides by 8: \[x = \frac{4}{8} = \frac{1}{2}\]

Step 5: Substitute x to Find y

Substitute \(x = \frac{1}{2}\) back into one of the original equations. Using the first equation: \[4\left(\frac{1}{2}\right) + 2y = 10\] Which simplifies to: \[2 + 2y = 10\] Solve for y: \[2y = 8\] \[y = 4\]

Step 6: Determine the Number of Solutions

Since we found a single solution \((x, y) = (\frac{1}{2}, 4)\), the system has exactly one solution.

Step 7: Classify the System

A system with exactly one solution is called 'consistent and independent'.

Key Concepts

Number of Solutions

Linear systems can have different numbers of solutions. Understanding these helps to classify the system.

• One Solution: If the two lines intersect at exactly one point, the system has a unique solution. This is known as having one solution.
• No Solution: If the lines are parallel and never intersect, the system has no solution. This is because there are no points that satisfy both equations at the same time.
• Infinite Solutions: If the two lines coincide, meaning they are the same line, then there are infinite solutions. Any point on the line is a solution.

To determine the number of solutions, you can analyze the coefficients of the equations or solve the system step-by-step, as shown in the exercise example.

Consistent and Independent Systems

Classifying a linear system involves terms like 'consistent' and 'independent'.

• Consistent: A system that has at least one solution is consistent. This means the equations do not contradict each other.
• Inconsistent: A system with no solutions is inconsistent. The equations represent parallel lines that never meet.
• Independent: If a system has exactly one solution, it is both consistent and independent. In this case, the equations represent lines that intersect at one unique point.
• Dependent: A system is dependent if it has infinitely many solutions. This happens when the equations describe the same line.

From the example, the system is consistent and independent because it has exactly one solution, \( (\frac{1}{2}, 4) \).

Linear Equations

Linear equations form the basis of linear systems. They are equations of the first degree, meaning the highest power of the variable is one. The general form is: \[ ax + by = c \].

• Equations in Two Variables: These are typically written as \[ ax + by = c \] where \( x \) and \( y \) are variables, and \( a \), \( b \), and \( c \) are constants.
• Slope-Intercept Form: Another common form is \[ y = mx + b \], where \( m \) is the slope of the line and \( b \) is the y-intercept.

When solving systems of linear equations, we use these forms to find points of intersection which represent the solutions. In the given exercise, adding the equations simplified the system, making it easy to solve for the variables.
Remember, understanding the structure of linear equations is essential for solving and classifying linear systems.