Math Problem Statement
Determine the x and y values of the following matrix similarities
[[-1 6x-1], [4x+5 3]] = [[-1 3y+2], [2x 3]]
Solution
To solve the given matrix similarity equation:
-1 & 6x-1 \\ 4x+5 & 3 \end{bmatrix} = \begin{bmatrix} -1 & 3y+2 \\ 2x & 3 \end{bmatrix}$$ We equate the corresponding elements of the matrices: 1. First row, first column: $$-1 = -1$$ (This equation is already satisfied and gives no information about $$x$$ or $$y$$). 2. First row, second column: $$6x - 1 = 3y + 2$$ 3. Second row, first column: $$4x + 5 = 2x$$ 4. Second row, second column: $$3 = 3$$ (This equation is already satisfied and gives no information about $$x$$ or $$y$$). Now, solve the equations: ### Step 1: Solve $$4x + 5 = 2x$$ $$4x + 5 = 2x$$ Subtract $$2x$$ from both sides: $$2x + 5 = 0$$ Subtract $$5$$ from both sides: $$2x = -5$$ Divide by $$2$$: $$x = -\frac{5}{2}$$ ### Step 2: Solve $$6x - 1 = 3y + 2$$ Substitute $$x = -\frac{5}{2}$$ into $$6x - 1 = 3y + 2$$: $$6\left(-\frac{5}{2}\right) - 1 = 3y + 2$$ Simplify: $$-15 - 1 = 3y + 2$$ $$-16 = 3y + 2$$ Subtract $$2$$ from both sides: $$-18 = 3y$$ Divide by $$3$$: $$y = -6$$ ### Final Values: $$x = -\frac{5}{2}, \quad y = -6$$ Let me know if you need further explanation or details! Here are 5 related questions you might explore: 1. How do matrix similarity transformations relate to eigenvalues? 2. What are the implications of equating elements in matrix similarity equations? 3. How do we find matrix inverses for 2x2 matrices, and when are they needed? 4. How would the process differ if the matrices were not equal but related by a transformation? 5. How can systems of equations from matrices be solved graphically? **Tip**: Always simplify individual equations to isolate variables for substitution effectively!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Matrix Equality
Systems of Linear Equations
Formulas
Equating corresponding elements of matrices
Theorems
Matrix Equality Theorem
Suitable Grade Level
Grades 9-12