Math Problem Statement
Now give the solution in terms of a and b to 2 4 1 1 x y = a b .
Solution
The given system of equations is represented in matrix form as:
2 & 4 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} a \\ b \end{pmatrix}$$ To solve for $$x$$ and $$y$$, we can express the system of equations as: 1. $$2x + 4y = a$$ 2. $$x + y = b$$ ### Step 1: Solve the second equation for $$x$$: $$x + y = b \quad \Rightarrow \quad x = b - y$$ ### Step 2: Substitute $$x = b - y$$ into the first equation: $$2(b - y) + 4y = a$$ $$2b - 2y + 4y = a$$ $$2b + 2y = a$$ $$2y = a - 2b$$ $$y = \frac{a - 2b}{2}$$ ### Step 3: Substitute $$y = \frac{a - 2b}{2}$$ into $$x = b - y$$: $$x = b - \frac{a - 2b}{2}$$ $$x = b - \frac{a}{2} + b$$ $$x = 2b - \frac{a}{2}$$ ### Final Solution: $$x = 2b - \frac{a}{2}, \quad y = \frac{a - 2b}{2}$$ Would you like further details or have any questions? --- Here are 5 related questions: 1. How do you invert a 2x2 matrix? 2. Can this method be extended to larger systems of equations? 3. What is the determinant of the matrix in this system? 4. How can you interpret the solution geometrically? 5. How does the system change if $$a = b$$? **Tip:** Always verify the solution by substituting it back into the original equations!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Matrix Representation
Formulas
2x + 4y = a
x + y = b
y = (a - 2b) / 2
x = 2b - a / 2
Theorems
Matrix Equation Solving
Substitution Method in Systems of Equations
Suitable Grade Level
Grades 10-12