Math Problem Statement
Determine the values of x and y from the following matrix equation: [4, 3x - y; 18, 6] = [4, 12; xy, 6].
Solution
The image contains a system of linear equations in matrix form that can be described as follows:
The matrix equation is given by:
4 & 3x - y \\ 18 & 6 \end{pmatrix} = \begin{pmatrix} 4 & 12 \\ xy & 6 \end{pmatrix}$$ We are tasked with determining the values of $$x$$ and $$y$$. ### Step-by-step Solution: 1. **Compare the elements of the matrices:** Since the two matrices are equal, corresponding elements must be equal. - First row, first column: $$4 = 4$$ (This is correct). - First row, second column: $$3x - y = 12$$. - Second row, first column: $$18 = xy$$. - Second row, second column: $$6 = 6$$ (This is also correct). 2. **Solve the system of equations:** Now, solve for $$x$$ and $$y$$ using the two equations: - $$3x - y = 12$$ (Equation 1) - $$xy = 18$$ (Equation 2) ### Solving the System: 1. From Equation 1: $$y = 3x - 12$$. 2. Substitute $$y = 3x - 12$$ into Equation 2: $$x(3x - 12) = 18$$ Expand the equation: $$3x^2 - 12x = 18$$ Rearrange into a standard quadratic equation: $$3x^2 - 12x - 18 = 0$$ Divide through by 3: $$x^2 - 4x - 6 = 0$$ 3. Solve the quadratic equation using the quadratic formula: $$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-6)}}{2(1)}$$ Simplifying: $$x = \frac{4 \pm \sqrt{16 + 24}}{2} = \frac{4 \pm \sqrt{40}}{2} = \frac{4 \pm 2\sqrt{10}}{2}$$ $$x = 2 \pm \sqrt{10}$$ Thus, $$x$$ can be $$x = 2 + \sqrt{10}$$ or $$x = 2 - \sqrt{10}$$. 4. Substitute these values of $$x$$ back into $$y = 3x - 12$$ to find $$y$$: - For $$x = 2 + \sqrt{10}$$: $$y = 3(2 + \sqrt{10}) - 12 = 6 + 3\sqrt{10} - 12 = 3\sqrt{10} - 6$$ - For $$x = 2 - \sqrt{10}$$: $$y = 3(2 - \sqrt{10}) - 12 = 6 - 3\sqrt{10} - 12 = -6 - 3\sqrt{10}$$ ### Final Answer: The solutions are: - $$x = 2 + \sqrt{10}$$, $$y = 3\sqrt{10} - 6$$ - $$x = 2 - \sqrt{10}$$, $$y = -6 - 3\sqrt{10}$$ Would you like more details on any part of this process? ### Related Questions: 1. How do you solve a system of linear equations using substitution? 2. What are the properties of matrix equality? 3. How can the quadratic formula be derived? 4. What does solving a quadratic equation geometrically look like? 5. What methods can be used to solve non-linear systems? ### Tip: Always check the solutions by substituting back into the original equations to ensure they satisfy both.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Matrix Equality
Quadratic Equations
Formulas
Quadratic equation: ax^2 + bx + c = 0
Theorems
Matrix equality theorem
Quadratic formula
Suitable Grade Level
Grades 10-12