Problem 56 - 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. $$ \left\\{\begin{array}{l} 3 x+4 y=12 \\ y=-3 x-1 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has one unique solution and it is consistent and independent.

Step 1 - Rewrite the equations

Rewrite the given system of equations in a standard format. The system is: 1) 3x + 4y = 12 2) y = -3x - 1

Step 2 - Express in standard form

Convert the second equation into standard form: From y = -3x - 1, multiply both sides by 3:3y = -9x - 3

Step 3 - Align coefficients for comparison

Now, align it with the standard form equation:1) 3x + 4y = 12 2) 9x + 3y = -3

Step 4 - Compare coefficients

Compare the coefficients of the two equations:For equation 1: 3x + 4y = 12For equation 2 (multiplied form): 9x + 3y = -3To directly compare, we observe the coefficients:First equation coefficient ratio: a1/a2 = 3/9 and b1/b2 = 4/3Thus, (a1/a2) ≠ (b1/b2)

Step 5 - Determine the consistency

Since the ratios of the coefficients are not equal, the system of equations represents two lines with different slopes. Therefore, they must intersect at one point.

Step 6 - Classify the system

A system of equations having exactly one intersection point is called 'consistent and independent'.

Key Concepts

Solving Linear Systems

Linear systems consist of two or more equations with the same set of variables. The primary goal is to find the values of these variables that satisfy all equations simultaneously. There are various methods to solve these systems, including:

• **Substitution:** Solve one equation for one variable and plug that back into the other equation.
• **Elimination:** Add or subtract equations to eliminate a variable, making it easier to solve for the remaining variables.
• **Graphing:** Plot both equations on the coordinate plane; their intersection(s) represent the solution(s).

In our problem, instead of graphing, we used the **elimination method** by converting the given equations into a comparable format. This helps us observe if there’s a common solution, simplifying the task significantly.

Classification of Systems

Systems of linear equations can generally be classified into three categories based on the number of solutions they have:

• **Consistent Independent:** Exactly one solution; the lines intersect at one point.
• **Consistent Dependent:** Infinite solutions; the lines are coincident (overlap each other).
• **Inconsistent:** No solution; the lines are parallel and never intersect.

By examining the coefficients of the variables as we did in the step-by-step solution and comparing their ratios, we can classify the system accordingly. When the ratios of the coefficients of the variables equal, we have parallel lines indicating inconsistency, but when they are different, it's either consistent independent or dependent.

Consistent Systems

A consistent system of linear equations implies that there is at least one solution that satisfies both equations. These systems are further subdivided into:

• **Consistent and Independent:** Here, the lines intersect at exactly one point, providing a unique solution. In our example, the system met this criterion as the ratios of coefficients for x and y were different, indicating the lines' intersection once.
• **Consistent and Dependent:** Lines overlap completely, indicating all points on one line are also on the other, giving infinitely many solutions. This happens when the ratios of the coefficients, as well as the constant terms, are proportional.

Consistent systems are quite typical in linear algebra and are crucial in solving real-world problems where making predictions or decisions based on some conditions is necessary.