Math Problem Statement
Solve the system of linear equations: y = x + 2 and y = -2x - 2.
Solution
This is a system of two linear equations:
y = x + 2 \\ y = -2x - 2 \end{cases}$$ To solve this system of equations, we can use either substitution or elimination. Let's solve it using substitution, as both equations are already solved for $$y$$. ### Step 1: Set the equations equal to each other Since both equations are equal to $$y$$, we can set the right-hand sides of the equations equal to each other: $$x + 2 = -2x - 2$$ ### Step 2: Solve for $$x$$ To solve for $$x$$, first add $$2x$$ to both sides: $$x + 2x + 2 = -2 \\ 3x + 2 = -2$$ Next, subtract 2 from both sides: $$3x = -4$$ Finally, divide by 3: $$x = -\frac{4}{3}$$ ### Step 3: Solve for $$y$$ Now that we have $$x = -\frac{4}{3}$$, we can substitute this value back into either of the original equations to find $$y$$. Let's use the first equation $$y = x + 2$$: $$y = -\frac{4}{3} + 2$$ Convert 2 to a fraction: $$y = -\frac{4}{3} + \frac{6}{3} = \frac{2}{3}$$ ### Final Answer: The solution to the system is: $$\left( x, y \right) = \left( -\frac{4}{3}, \frac{2}{3} \right)$$ Would you like more details or have any questions about the solution? Here are five related questions that you might find interesting: 1. How would you graph these two equations on the coordinate plane? 2. What is the significance of the point where these two lines intersect? 3. How can the elimination method be applied to this system? 4. What does the solution represent in real-world applications? 5. Can you find another pair of linear equations with the same solution? **Tip:** When solving a system of equations, always check your solution by substituting the values back into the original equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Systems of Equations
Formulas
y = mx + b
Substitution method in solving systems of equations
Theorems
Equality of linear functions
Suitable Grade Level
Grades 9-10