In Exercises 1–18, solve each system by the substitution method. x+y=1, (x-1)^2+(y+2)^2=10

Verified step by step guidance

1. Step 1: The first step in solving this system of equations by substitution is to solve one of the equations for one variable. In this case, the first equation, x + y = 1, is already solved for x. We can rewrite it as x = 1 - y.
2. Step 2: Substitute x = 1 - y into the second equation. This gives us: (1 - y - 1)^2 + (y + 2)^2 = 10. Simplify this to get: y^2 + (y + 2)^2 = 10.
3. Step 3: Now, you have a quadratic equation in terms of y. Simplify the equation by expanding the squares and combining like terms. This will give you a quadratic equation in the standard form ax^2 + bx + c = 0.
4. Step 4: Solve the quadratic equation for y. You can do this by factoring, completing the square, or using the quadratic formula, depending on what the equation looks like.
5. Step 5: Once you have the solutions for y, substitute them back into the equation x = 1 - y to find the corresponding x values. These pairs of x and y values are the solutions to the system of equations.

Key Concepts

Substitution Method

The substitution method is a technique for solving systems of equations where one equation is solved for one variable in terms of the other. This expression is then substituted into the second equation, allowing for the solution of one variable. Once one variable is found, it can be substituted back to find the other variable, leading to a complete solution of the system.

Linear Equations

Linear equations are mathematical statements that describe a straight line when graphed. They typically take the form Ax + By = C, where A, B, and C are constants. In the given system, the equation x + y = 1 is linear, representing a line in the coordinate plane, which can intersect with other equations to find solutions.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, generally expressed in the form ax^2 + bx + c = 0. The equation (x-1)^2 + (y+2)^2 = 10 represents a circle in the coordinate plane, which is a geometric representation of a quadratic equation. Understanding the properties of quadratic equations is essential for analyzing their intersections with linear equations in a system.