Solve the System of Equations x^2+y^2=10 y=2x-5
The problem presents a system of equations that consists of two equations with two variables, x and y. The first equation, x^2 + y^2 = 10, represents a circle with a certain radius centered at the origin on a coordinate plane. The second equation, y = 2x - 5, represents a straight line. The question asks to find the point(s) where the circle and the line intersect by solving this system simultaneously.
Substitute
Substitute
Expand and simplify the equation.
Expand
Expand each term individually.
Rewrite
Use the FOIL Method to expand
Distribute
Combine like terms.
Simplify and combine like terms.
Combine
Solve for
Subtract
Factor the quadratic equation.
Factor out the greatest common factor, which is
Factor
Factor the trinomial
Find two numbers that multiply to
Set each factor equal to zero and solve for
Solve
Solve
The solutions for
Substitute
Simplify to find
Substitute
Simplify to find
The solution set is the pair of
The solution can be presented in different formats.
Point Form:
The problem involves solving a system of equations, one of which is a circle equation
Substitution: Replace
Simplification: Simplify the equation by expanding and combining like terms.
Factoring: Factor the resulting quadratic equation to find the values of
Solving Quadratic Equations: Set each factor equal to zero and solve for
Back Substitution: Substitute the found values of
Solution Set: The solution to the system of equations is the set of ordered pairs
The FOIL Method (First, Outer, Inner, Last) is used to expand binomials, and factoring involves finding two numbers that multiply to the constant term and add to the coefficient of the middle term in a quadratic equation. The distributive property is applied when expanding expressions, and the zero-product property is used to find the solutions of a factored equation.