Math Problem Statement
On a piece of paper, graph the system of equations. Then determine which answer choice matches the graph you drew and identify the solution to the system. y = -2x - 1, y = x + 5.
Solution
The question asks to graph the system of equations:
Steps to Solve:
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Graph the first equation ():
- Y-intercept: The equation is in slope-intercept form (), where is the y-intercept. Here, , so the graph passes through .
- Slope: The slope () is , meaning for every 1 unit increase in , decreases by 2 units. Another point can be calculated as follows:
- Starting at : Increase by 1 → , .
- Points: , .
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Graph the second equation ():
- Y-intercept: Here, , so the graph passes through .
- Slope: The slope () is , meaning for every 1 unit increase in , also increases by 1. Another point can be calculated:
- Starting at : Increase by 1 → , .
- Points: , .
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Find the Intersection Point:
- Solve the equations simultaneously: Combine like terms: Solve for : Substitute into one of the equations, say : Intersection point: .
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Conclusion:
- The graphs intersect at , which is the solution to the system of equations.
Let me know if you would like a visual graph of this system or further clarification!
Related Questions:
- How can you verify that satisfies both equations?
- What happens if the slopes of the two lines are equal?
- How can you determine the solution without graphing?
- How does changing the slope or y-intercept affect the graph of a line?
- What are other methods for solving systems of linear equations?
Tip: Always label the axes and use a ruler for accuracy when graphing linear equations!
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Math Problem Analysis
Mathematical Concepts
Algebra
Graphing Systems of Linear Equations
Intersection of Lines
Formulas
Slope-Intercept Form: y = mx + b
Theorems
Point of Intersection Theorem for Linear Systems
Suitable Grade Level
Grades 8-10