Math Problem Statement
Solution
The problem involves graphing two linear equations and on a coordinate plane and determining their point of intersection. Let me guide you step by step:
Step 1: Equations to graph
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Equation 1:
- This is a line with a slope of 2 and a y-intercept of 8. Start by plotting the point (0, 8), then use the slope to find additional points, such as:
- From (0, 8), go up 2 units and right 1 unit to reach (1, 10).
- Alternatively, go down 2 units and left 1 unit to reach (-1, 6).
- This is a line with a slope of 2 and a y-intercept of 8. Start by plotting the point (0, 8), then use the slope to find additional points, such as:
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Equation 2:
- This is a line with a slope of and a y-intercept of -4. Start by plotting the point (0, -4), then use the slope :
- From (0, -4), go up 1 unit and right 2 units to reach (2, -3).
- Alternatively, go down 1 unit and left 2 units to reach (-2, -5).
- This is a line with a slope of and a y-intercept of -4. Start by plotting the point (0, -4), then use the slope :
Step 2: Intersection point
To find the intersection, solve the equations and algebraically:
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Set the equations equal:
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Eliminate the fraction by multiplying the entire equation by 2:
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Simplify and solve for :
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Substitute into one of the equations (e.g., ) to find :
Thus, the point of intersection is .
Step 3: Graph the lines
- Plot the lines for and .
- Verify they intersect at .
Would you like further clarification or assistance with graphing? Let me know!
Follow-Up Questions:
- What is the significance of the slopes of the two lines?
- How do you confirm the intersection point visually from the graph?
- Can two parallel lines have an intersection point? Why or why not?
- What happens if both equations are the same (coincident lines)?
- How would you graph a vertical line (e.g., )?
Tip: Always double-check your graph to ensure it reflects the calculated intersection point.
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Math Problem Analysis
Mathematical Concepts
Algebra
Graphing Linear Equations
Finding Intersection Points
Formulas
Slope-intercept form y = mx + b
Solving linear equations algebraically
Theorems
Linear Equation Theory
Suitable Grade Level
Grades 8-10
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