Math Problem Statement
Solve the system of linear equations: -x - 3y = -6 and 2y + x = 6.
Solution
The system of linear equations in the image is:
-x - 3y = -6 \\ 2y + x = 6 \end{cases}$$ ### Step 1: Solve for $$x$$ in terms of $$y$$ from the second equation From $$2y + x = 6$$, we isolate $$x$$: $$x = 6 - 2y$$ ### Step 2: Substitute $$x = 6 - 2y$$ into the first equation Substituting into $$-x - 3y = -6$$: $$-(6 - 2y) - 3y = -6$$ Simplify: $$-6 + 2y - 3y = -6$$ $$-6 - y = -6$$ $$-y = 0$$ $$y = 0$$ ### Step 3: Substitute $$y = 0$$ back into $$x = 6 - 2y$$ Substitute $$y = 0$$ into $$x = 6 - 2y$$: $$x = 6 - 2(0)$$ $$x = 6$$ ### Solution: The solution to the system of equations is: $$x = 6, \, y = 0$$ Would you like a detailed explanation of the substitution method or to explore graphing these equations? Here are some related questions: 1. What are other methods to solve systems of equations like this one, such as elimination or graphing? 2. How can we verify that the solution $$(6, 0)$$ satisfies both equations? 3. Can this method be extended to systems with more variables (e.g., three equations)? 4. What happens if a system has no solutions or infinitely many solutions? How can we identify such cases? 5. How does the solution change if we rewrite the equations differently? **Tip:** Always double-check your solution by substituting the values 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
Linear equations
Substitution method
System of equations
Formulas
Solving for a variable: ax + by = c
Theorems
Basic linear algebra principles
Suitable Grade Level
Grades 8-10