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

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Describe the graph of a linear equation

Full solution

Q.

y=x-2

and

x^2+y^2=52

Identify Equation Types: Identify the type of the equations given. $\newline$ The first equation $y = x - 2$ is a linear equation, and the second equation $x^2 + y^2 = 52$ is a circle with radius $\sqrt{52}$ centered at the origin.
Substitute $y$ Expression: Substitute the expression for $y$ from the first equation into the second equation. $\newline$ 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. $\newline$ 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. $\newline$ Dividing by $2$ gives $x^2 - 2x - 24 = 0$ .
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ 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. $\newline$ 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. $\newline$ For $(6, 4)$ , we check $6^2 + 4^2 = 52$ , which simplifies to $36 + 16 = 52$ , which is true. $\newline$ For $(-4, -6)$ , we check $(-4)^2 + (-6)^2 = 52$ , which simplifies to $16 + 36 = 52$ , which is also true.

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