In Exercises 1–18, solve each system by the substitution method. x+y=1, x^2+xy-y^2=-5

Verified step by step guidance

1. Step 1: From the first equation, we can express x in terms of y. This is done by subtracting y from both sides of the equation. So, we get x = 1 - y.
2. Step 2: Substitute x = 1 - y into the second equation. This will give us a quadratic equation in terms of y. The equation will be (1 - y)^2 + (1 - y)y - y^2 = -5.
3. Step 3: Simplify the equation obtained in step 2. This will give us a quadratic equation in standard form, which is ay^2 + by + c = 0.
4. Step 4: Solve the quadratic equation obtained in step 3 for y. This can be done by using the quadratic formula, y = [-b ± sqrt(b^2 - 4ac)] / (2a).
5. Step 5: Substitute the solutions for y obtained in step 4 back into the equation x = 1 - y to find the corresponding x values. This will give us the solutions to the system of equations.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This method simplifies the system, allowing for easier solving of the equations, especially when one equation is linear.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed. They can be written in the form Ax + By = C, where A, B, and C are constants. In the given system, the equation x + y = 1 is linear, which allows for straightforward manipulation and substitution.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax^2 + bx + c = 0. The equation x^2 + xy - y^2 = -5 contains both linear and quadratic terms, making it essential to recognize how to handle these types of equations when substituting values from one equation into another.