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16 (a) On the grid, draw the graph of $x^2 + y^2 = 12.25$ (b) Hence find estimates for the solutions of the simultaneous equations $x^2 + y^2 = 12.25$ $2x + y = 1$ - Edexcel - GCSE Maths - Question 17 - 2018 - Paper 2
Question 17
16 (a) On the grid, draw the graph of $x^2 + y^2 = 12.25$ (b) Hence find estimates for the solutions of the simultaneous equations $x^2 + y^2 = 12.25$ $2x + y = 1$
Worked Solution & Example Answer:16 (a) On the grid, draw the graph of $x^2 + y^2 = 12.25$ (b) Hence find estimates for the solutions of the simultaneous equations $x^2 + y^2 = 12.25$ $2x + y = 1$ - Edexcel - GCSE Maths - Question 17 - 2018 - Paper 2
Step 1
Draw the graph of $x^2 + y^2 = 12.25$
Answer
To draw the graph of the equation , we recognize that this represents a circle centered at the origin (0,0) with a radius of rac{12.25}{ ext{radius}^2} = 3.5. To plot the graph:
- Calculate the center and radius. We know it’s centered at the origin, and the radius is approximately 3.5.
- Plot points on the circle. For example, where , , and where , .
- Connect these points smoothly to form a circle.
Step 2
Estimate solutions for $x^2 + y^2 = 12.25$ and $2x + y = 1$
Answer
To find estimates for the solutions of the simultaneous equations, we need to substitute values from the linear equation into the circular one. From the equation , we can express as:
.
Now we substitute this into the circle's equation:
Expanding this gives:
Simplifying further:
5x^2 - 4x - 11.25 = 0$$ Using the quadratic formula, we can find the x-values and then substitute back into $y = 1 - 2x$ to get corresponding y-values. The estimated intersection points can be represented as pairs on the graph.