Math Problem Statement

Classify and solve systems of linear equations as provided in the image.

Solution

Let's work through each of the cases in this problem.

Caso 1: Clasificación de los sistemas de ecuaciones lineales

We need to classify each system as:

  1. Compatible determinado (a system with a unique solution),
  2. Compatible indeterminado (a system with infinitely many solutions), or
  3. Incompatible (a system with no solution).

Analyzing Each System:

  1. System A: 3x - 4y = 11 \\ -1.5x + 2y = -5.5 \\ \end{cases}$$ If we manipulate this system, we can see that multiplying the second equation by 2 gives: $$\begin{cases} 3x - 4y = 11 \\ -3x + 4y = -11 \\ \end{cases}$$ Adding both equations results in $$0 = 0$$, indicating that this system is consistent and has infinitely many solutions. Therefore, it is **Compatible Indeterminado** (II).
  2. System B: 5x + 3y = 21 \\ 4x + 7y = 12 \\ \end{cases}$$ This system does not yield parallel or coincident lines after simplification. Therefore, it has a unique solution and is **Compatible Determinado** (I).
  3. System C: 7x - 5y = 23 \\ 0.7x - 0.5y = 1 \\ \end{cases}$$ Multiplying the second equation by 10 gives: $$7x - 5y = 10 \\$$ Since the left sides are identical but the right sides are different, this system is inconsistent, meaning it has no solution. It is **Incompatible** (III).

Final Classification for Caso 1:

The correct answer is: La relacioˊn es: B,I;A,II;C,III\text{La relación es: } \text{B,I;A,II;C,III}

Caso 2: Resolving the System of Equations

We are given:

2x - 5y = -13 \\ 3x + 4y = 61 \\ \end{cases}$$ To solve this system, we can use either substitution or elimination. Here, we'll use the elimination method. 1. Multiply the first equation by 3 and the second equation by 2 to align the $$x$$ terms: $$\begin{cases} 6x - 15y = -39 \\ 6x + 8y = 122 \\ \end{cases}$$ 2. Subtract the first equation from the second: $$23y = 161 \Rightarrow y = 7$$ 3. Substitute $$y = 7$$ back into the first equation: $$2x - 5(7) = -13 \Rightarrow 2x - 35 = -13 \Rightarrow 2x = 22 \Rightarrow x = 11$$ #### Final Solution for Caso 2: $$\text{El c.s. = } \{ (11, 7) \}$$ --- ### **Summary** - **Caso 1 Answer**: B,I;A,II;C,III - **Caso 2 Answer**: $$\{ (11, 7) \}$$ --- Would you like further details on any part of the solution? ### Additional Questions to Explore: 1. What methods can be used to identify if a system is compatible or incompatible? 2. How can elimination and substitution methods be applied to solve linear systems? 3. What does it mean for a system to be "indeterminate" in terms of solution types? 4. How can we check for parallel or coincident lines in linear systems? 5. How can matrix methods help in solving systems of linear equations? ### Tip: When classifying linear systems, observe if manipulating one equation can make it identical to another—this often reveals compatibility or incompatibility quickly.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Equation Classification
Consistent and Inconsistent Systems

Formulas

Elimination method
Substitution method
Matrix representation for linear systems

Theorems

Theory of Consistent and Inconsistent Systems
Linear Independence in Equation Systems

Suitable Grade Level

Grades 10-12

Related Recommendation

Solving Systems of Linear Equations with Substitution and Elimination

Solving Systems of Equations: Methods and Examples

Solving Systems of Equations, Inequalities, and Matrix Equations

Solving Systems of Linear Equations: Are Systems A and B Equivalent?

Solving Systems of Linear Equations Step-by-Step