Math Problem Statement

Use the echelon method to solve the following system of two equations in two unknowns. Check your answer: 4x - 3y = -1, -8x + 6y = 2.

Solution

Let's solve the system of equations using the echelon method:

The given system is:

4x - 3y = -1 \tag{1} -8x + 6y = 2 \tag{2}

Step 1: Create an augmented matrix for the system:

4 & -3 & | & -1 \\ -8 & 6 & | & 2 \end{pmatrix}$$ ### Step 2: Apply row operations to achieve echelon form. First, we can divide the first row by 4 to make the leading coefficient of $$x$$ in the first equation equal to 1: $$\begin{pmatrix} 1 & -\frac{3}{4} & | & -\frac{1}{4} \\ -8 & 6 & | & 2 \end{pmatrix}$$ Next, we'll eliminate the $$x$$-term from the second row by adding 8 times the first row to the second row: $$R_2 = R_2 + 8R_1$$ This gives us: $$\begin{pmatrix} 1 & -\frac{3}{4} & | & -\frac{1}{4} \\ 0 & 0 & | & 0 \end{pmatrix}$$ ### Step 3: Interpret the echelon form. The second row indicates $$0 = 0$$, which is always true. Therefore, the system does not provide any new information from the second equation. The first row represents: $$x - \frac{3}{4}y = -\frac{1}{4}$$ which can be rewritten as: $$x = \frac{3}{4}y - \frac{1}{4}$$ ### Step 4: Solution. Since the second row is trivially true, there are infinitely many solutions. The solution can be expressed in parametric form as: $$x = \frac{3}{4}y - \frac{1}{4}, \quad y = y$$ where $$y$$ is any real number. Thus, the answer is **option B**: There are infinitely many solutions, and the general solution is $$(\frac{3}{4}y - \frac{1}{4}, y)$$, where $$y$$ is any real number. --- Do you have any questions or need further details? Here are 5 related questions: 1. What is the significance of echelon form in solving systems of equations? 2. How do you identify when a system of linear equations has no solution? 3. How can you express the solution of a system with infinitely many solutions parametrically? 4. What other methods can be used to solve systems of linear equations besides the echelon method? 5. Why does a row of zeros in a matrix correspond to an identity like $$0 = 0$$? **Tip:** When dealing with systems of equations, always check if you can simplify one of the equations to make row operations easier.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Echelon Method

Formulas

Augmented Matrix
Row Operations

Theorems

Echelon Form
Parametric Solution

Suitable Grade Level

Grades 10-12

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