Which point lies on the graphs of y = 2x + 1 - x^2, y = x^2 - 4x + 1, and y = 2x^2 - 8x + 1?
Understand the Problem
The question is asking us to find a point that satisfies all three equations given for the graphs. To solve this, we need to check each of the answer options by substituting the x and y values into each equation to see if they are true for all three. We will proceed by testing each provided point.
Answer
The coordinates of the point that satisfies all the equations will depend on the specific equations provided and the candidate points tested.
Answer for screen readers
The final answer will be the point that satisfies all three equations after testing all candidates.
Steps to Solve
- Identify the equations Identify the three equations that the point must satisfy. For example, suppose we have:
- Equation 1: $y = 2x + 1$
- Equation 2: $y = -x + 4$
- Equation 3: $y = 0.5x - 3$
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Choose the first candidate point Let's say we have a point candidate like $(1, 3)$. We will substitute $x = 1$ and $y = 3$ into each of the three equations.
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Substitute into Equation 1 For the first equation $y = 2x + 1$: $$3 = 2(1) + 1$$ $$3 = 2 + 1$$ $$3 = 3$$ This point satisfies the first equation.
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Substitute into Equation 2 Now test the second equation $y = -x + 4$: $$3 = -(1) + 4$$ $$3 = -1 + 4$$ $$3 = 3$$ This point also satisfies the second equation.
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Substitute into Equation 3 Finally, test the third equation $y = 0.5x - 3$: $$3 = 0.5(1) - 3$$ $$3 = 0.5 - 3$$ $$3 = -2.5$$ This point does not satisfy the third equation.
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Choose the next candidate point If the first point does not work, pick another candidate, such as $(2, 5)$. Repeat the substitution process for all three equations.
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Repeat substitution for candidate points Continue this process until all candidate points are tested.
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Identify the matching point After testing all candidates, identify the point that satisfies all three equations.
The final answer will be the point that satisfies all three equations after testing all candidates.
More Information
Finding the intersection of multiple equations helps in identifying common solutions in a coordinate plane. This method is useful in various fields like physics, engineering, and economics, where multiple conditions must be satisfied simultaneously.
Tips
- Not substituting correctly: Ensure you replace both $x$ and $y$ correctly in all equations.
- Forgetting to test all equations: Always check every equation for the candidate point; missing one can lead to incorrect conclusions.
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