Problem 32 - Algebra Form and Function

Question

Without solving the equations, decide how many solutions the system has. $$ \left\\{\begin{array}{r} 4 x-y=2 \\ 12 x-3 y=2 \end{array}\right. $$

Step-by-Step Solution

Answer

Equations: 4x - y = 8 12x - 3y = 24 Answer: The given system of linear equations has infinite solutions.

Step 1: Identify the coefficients of the given linear equations

First, we need to identify the coefficients of the two given linear equations to compare them. For the first equation, the coefficients are (4, -1) and for the second equation, the coefficients are (12, -3).

Step 2: Assess the ratio between the coefficients

To check if these two equations are equivalent, parallel, or neither, we will now compare the ratios of their coefficients. For this, we will divide the coefficient of x in the second equation by the coefficient of x in the first equation, and the coefficient of y in the second equation by the coefficient of y in the first equation: \[\frac{12}{4}=\frac{-3}{-1}\]

Step 3: Determine the relationship between the coefficients

Since the ratio between both coefficients is equal (the result of the division is 3 for both coefficients), the equations are equivalent. They have the same slope and different constant terms, which means the two lines represented by the equations overlap.

Step 4: Identify the number of solutions

In the case of equivalent equations, there are infinite solutions, as the two lines overlap in all points. Therefore, our conclusion for this system of linear equations is that it has infinite solutions.

Key Concepts

Understanding Systems of Equations

In mathematics, a system of equations is essentially a set of two or more equations that share two or more unknowns. We take on systems of equations when we need to find the solutions that work for all equations simultaneously. Consider scenarios in real life, like balancing different financial budgets or mixing chemicals in a laboratory; systems of equations perfectly model these situations as they represent multiple conditions that must all be satisfied simultaneously.

• Each equation in the system provides a constraint on the variables involved.
• The goal is to find values for each variable that satisfy every equation in the system.
• Graphically, each equation can be viewed as a line in a coordinate plane. The solution to the system is where these lines intersect.

In our original example, the system consists of two linear equations, and we are tasked with determining how many solutions there are without actually solving the system. Identifying characteristics like the number and type of solutions help us understand the relationship between the equations.

Finding the Solution of Linear Systems

A solution of a system of linear equations refers to the set of values for the variables that satisfies all equations in the system at once. There are several possible outcomes for these solutions:

• One Solution: This occurs when the lines representing the equations intersect at exactly one point. It suggests the equations are independent and consistent.
• No Solution: When the lines are parallel and do not intersect at any point, the system is inconsistent. Thus, there are no solutions.
• Infinite Solutions: This happens when the lines overlap completely, meaning the equations are dependent, thus consistent.

In our problem, by assessing the coefficients and finding that they have the same ratio, we can deduce that the equations are equivalent, leading to infinite intersecting points. Therefore, the given system has infinite solutions, demonstrating the lines completely overlap on the graph.

Grasping Equivalent Equations

Equivalent equations are equations that have the same set of solutions. In the realm of linear equations, two equations are equivalent if they represent the same line graphically. This means they share identical slopes and constant terms. Having equivalent equations in a system is crucial because:

• The equations describe the same relationship between variables.
• Graphically, they will plot as one line overlapping itself entirely.
• It offers insights into the nature of solutions (infinite in this case).

In the provided exercise, after comparing the equations' coefficients, it was determined that they maintain a consistent ratio, confirming their equivalence. They illustrate that under different forms or scaling factors, the equations fundamentally describe the same line, hence having infinite solutions. This understanding helps in effortlessly deducing outcomes without explicit calculations.