y=x-2 and x2+y2=52

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y=x-2 and x2+y2=52

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Identify Equation Types: Identify the type of the equations given. The first equation y = x - 2 is a linear equation, and the second equation x^2 + y^2 = 52 is a circle with radius √52 centered at the origin.Substitute y Expression: Substitute the expression for y from the first equation into the second equation. By substituting y = x - 2 into x^2 + y^2 = 52, we get x^2 + (x - 2)^2 = 52.Expand and Simplify: Expand and simplify the equation. Expanding (x - 2)^2 gives x^2 - 4x + 4, so the equation becomes x^2 + x^2 - 4x + 4 = 52.Combine Terms and Solve: Combine like terms and solve for x. Combining x^2 terms gives 2x^2 - 4x + 4 = 52. Subtract 52 from both sides to get 2x^2 - 4x - 48 = 0.Divide and Simplify: Divide the entire equation by 2 to simplify. Dividing by 2 gives x^2 - 2x - 24 = 0.Factor Quadratic Equation: Factor the quadratic equation. The equation factors to (x - 6)(x + 4) = 0.Find x Values: Find the values of x. Setting each factor equal to zero gives us x - 6 = 0 or x + 4 = 0, so x = 6 or x = -4.Substitute for y: Substitute the values of x back into the first equation to find the corresponding y values. For x = 6, y = 6 - 2 = 4. For x = -4, y = -4 - 2 = -6.Check Solutions: Check both pairs (x, y) in the second equation to ensure they satisfy it. For (6, 4), we check 6^2 + 4^2 = 52, which simplifies to 36 + 16 = 52, which is true. For (-4, -6), we check (-4)^2 + (-6)^2 = 52, which simplifies to 16 + 36 = 52, which is also true.

$y=x-2$ and $x^2+y^2=52$

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y=x−2y=x-2y=x−2 and x2+y2=52x^2+y^2=52x2+y2=52

$y=x-2$ and $x^2+y^2=52$