Math Problem Statement
Solve the system of equations: y = 2x + 2 and y = -7x - 1
Solution
The system of equations you've uploaded is:
y = 2x + 2 \\ y = -7x - 1 \end{cases}$$ We can solve this system using substitution, since both equations are already solved for $$y$$. ### Step-by-step solution: 1. Set the right-hand sides of both equations equal to each other since they both equal $$y$$: $$2x + 2 = -7x - 1$$ 2. Solve for $$x$$: - First, add $$7x$$ to both sides: $$2x + 7x + 2 = -1$$ $$9x + 2 = -1$$ - Then, subtract 2 from both sides: $$9x = -3$$ - Finally, divide by 9: $$x = -\frac{1}{3}$$ 3. Substitute $$x = -\frac{1}{3}$$ into either of the original equations to find $$y$$. Using $$y = 2x + 2$$: $$y = 2\left(-\frac{1}{3}\right) + 2$$ $$y = -\frac{2}{3} + 2$$ $$y = -\frac{2}{3} + \frac{6}{3} = \frac{4}{3}$$ ### Solution: The solution to the system is: $$x = -\frac{1}{3}, \quad y = \frac{4}{3}$$ Would you like further details on the method used, or have any other questions? Here are 5 related questions to deepen understanding: 1. How would you solve a system of equations using the elimination method? 2. What would happen if both equations had the same slope? 3. How do you interpret the solution geometrically on a graph? 4. Can systems of equations have more than one solution? Why or why not? 5. How do you solve a system of equations with more than two variables? **Tip**: When solving systems of equations, always check the solution by substituting back into both original equations to ensure consistency.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Systems of Equations
Formulas
y = 2x + 2
y = -7x - 1
Theorems
Substitution method for solving systems of equations
Suitable Grade Level
Grades 8-10