Math Problem Statement
Solve the system of equations: ax + by = (a - b), bx - ay = (a + b)
Solution
The system of equations in the image is:
1) & \quad ax + by = (a - b) \\ 2) & \quad bx - ay = (a + b) \end{aligned}$$ We need to solve this system of equations for $$x$$ and $$y$$. I'll solve it step by step: ### Step 1: Label the equations $$(1) \quad ax + by = a - b$$ $$(2) \quad bx - ay = a + b$$ ### Step 2: Eliminate one variable We can use the method of elimination. First, multiply both equations by constants so that we can eliminate either $$x$$ or $$y$$. - Multiply equation (1) by $$b$$: $$b(ax + by) = b(a - b) \implies abx + b^2y = ab - b^2 \quad (3)$$ - Multiply equation (2) by $$a$$: $$a(bx - ay) = a(a + b) \implies abx - a^2y = a^2 + ab \quad (4)$$ ### Step 3: Subtract (4) from (3) Now subtract equation (4) from equation (3): $$(abx + b^2y) - (abx - a^2y) = (ab - b^2) - (a^2 + ab)$$ Simplify the left and right-hand sides: $$b^2y + a^2y = -b^2 - a^2$$ Factor out $$y$$ on the left: $$(a^2 + b^2)y = -(a^2 + b^2)$$ ### Step 4: Solve for $$y$$ If $$a^2 + b^2 \neq 0$$, we can divide both sides by $$a^2 + b^2$$: $$y = -1$$ ### Step 5: Substitute $$y = -1$$ into one of the original equations Substitute $$y = -1$$ into equation (1): $$ax + b(-1) = a - b$$ $$ax - b = a - b$$ Add $$b$$ to both sides: $$ax = a$$ Solve for $$x$$: $$x = 1$$ ### Final Answer: The solution to the system is: $$x = 1, \quad y = -1$$ Let me know if you'd like any clarifications or more details! Here are some related questions: 1. Can we solve this system using substitution instead of elimination? 2. What happens if $$a^2 + b^2 = 0$$? Would the system still have a solution? 3. How does this system change if we swap the terms on the right-hand side of the equations? 4. Can you graph the system of equations and interpret the solution geometrically? 5. What happens if $$a = b$$? Does the solution change? **Tip**: In systems of equations, elimination is often helpful when coefficients are already aligned for one variable, while substitution is easier when one equation is already solved for one variable.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Elimination Method
Formulas
ax + by = a - b
bx - ay = a + b
Theorems
Elimination method for solving systems of linear equations
Linear equations theorem
Suitable Grade Level
Grades 9-12
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