Math Problem Statement
Graph the system of equations and write its solution: y = -3x - 3 and 6x + 2y = -2.
Solution
The system of equations given in the image is:
y = -3x - 3 \\ 6x + 2y = -2 \end{cases}$$ We need to graph the system and determine its solution. ### Step 1: Graph the first equation $$y = -3x - 3$$ This is already in slope-intercept form $$y = mx + b$$, where: - $$m = -3$$ (the slope), - $$b = -3$$ (the y-intercept). - To plot: Start at $$(0, -3)$$, then use the slope of $$-3$$ to move down 3 units and right 1 unit. ### Step 2: Rearrange the second equation $$6x + 2y = -2$$ Solve for $$y$$ to get it in slope-intercept form: $$6x + 2y = -2$$ $$2y = -6x - 2$$ $$y = -3x - 1$$ This equation also has: - $$m = -3$$, - $$b = -1$$ (the y-intercept). - To plot: Start at $$(0, -1)$$, and again use the slope $$-3$$ to move down 3 units and right 1 unit. ### Step 3: Analyze the graphs Both equations have the same slope $$m = -3$$, meaning the lines are parallel. Since their y-intercepts are different ($$-3$$ and $$-1$$), these lines will never intersect. ### Conclusion The system has no solution because the lines are parallel. Do you need further details or any other clarifications? Here are 5 related questions to explore the topic further: 1. How do you determine the slope and y-intercept from an equation in slope-intercept form? 2. What is the graphical interpretation of parallel lines in a system of equations? 3. How can you tell if a system of equations has infinitely many solutions? 4. What method can be used to solve a system of equations algebraically instead of graphically? 5. How would you handle a system of equations where one or both equations are not in slope-intercept form? **Tip:** When two lines have the same slope but different y-intercepts, they are always parallel and will never intersect, leading to no solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Graphing Linear Equations
Formulas
y = mx + b (Slope-Intercept Form)
Rearranging Linear Equations
Theorems
Parallel Lines Theorem
Suitable Grade Level
Grades 8-10
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