In Exercises 1–18, solve each system by the substitution method. y^2=x^2-9, 2y=x-3

Verified step by step guidance

1. Step 1: Solve the second equation for one variable in terms of the other. From the equation 2y = x - 3, express x as x = 2y + 3.
2. Step 2: Substitute the expression for x from Step 1 into the first equation. Replace x in the equation y^2 = x^2 - 9 with 2y + 3 to get y^2 = (2y + 3)^2 - 9.
3. Step 3: Expand the squared term in the equation from Step 2. Expand (2y + 3)^2 to get 4y^2 + 12y + 9.
4. Step 4: Simplify the equation by combining like terms. Substitute 4y^2 + 12y + 9 for (2y + 3)^2 in the equation, then simplify y^2 = 4y^2 + 12y + 9 - 9 to get y^2 = 4y^2 + 12y.
5. Step 5: Rearrange the equation to set it to zero and solve for y. Rearrange the equation to 0 = 3y^2 + 12y, then factor out the common term and solve for y.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of the other. In this method, you solve one equation for one variable and then substitute that expression into the other equation. This allows you to reduce the system to a single equation with one variable, making it easier to find the solution.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. In the context of the given question, the equation y^2 = x^2 - 9 can be rearranged to form a quadratic equation in terms of x or y. Understanding how to manipulate and solve quadratic equations is essential for finding the solutions to the system.

Graphing Systems of Equations

Graphing systems of equations involves plotting each equation on the same coordinate plane to visually identify points of intersection, which represent the solutions. In this case, the equations y^2 = x^2 - 9 and 2y = x - 3 can be graphed to observe how they interact. Understanding the graphical representation helps in verifying the solutions obtained through algebraic methods.