In Exercises 1–18, solve each system by the substitution method. {y2=x2−92y=x−3\begin{cases} $$y^2$$ = $$x^2 - 9$$ \\ 2y = x - 3 \end{cases}{y2=x2−92y=x−3

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 11

Chapter 6, Problem 11

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\)y^2 = x^2 - 9 \\2y = x - 3\(\end{cases}$

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Start with the given system of equations: $y^2 = x^2 - 9$ and $2y = x - 3$.

From the second equation, solve for $x$ in terms of $y$: $2y = x - 3$ implies $x = 2y + 3$.

Substitute the expression for $x$ from step 2 into the first equation: $y^2 = (2y + 3)^2 - 9$.

Expand the right side: $(2y + 3)^2 = 4y^2 + 12y + 9$, so the equation becomes $y^2 = 4y^2 + 12y + 9 - 9$.

Simplify and rearrange the equation to isolate terms and form a quadratic equation in $y$: $y^2 = 4y^2 + 12y$, then bring all terms to one side to solve for $y$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one equation is already solved or easily solvable for one variable.

Solving Quadratic Equations

Quadratic equations involve variables raised to the second power and can have two solutions. When substituting, the resulting equation may be quadratic, requiring methods like factoring, completing the square, or the quadratic formula to find solutions. Understanding how to solve quadratics is essential for finding all possible solutions.

Checking Solutions in Systems

After finding potential solutions, it is important to verify them by substituting back into the original equations. This ensures that the solutions satisfy both equations, especially since squaring or other operations can introduce extraneous solutions. Checking helps confirm the accuracy and validity of the answers.

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In Exercises 5–18, solve each system by the substitution method. 5x + 2y = 0 x - 3y = 0

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Solve each system in Exercises 12–13. The is a piecewise function

In Exercises 1–18, solve each system by the substitution method. {y2=x2−92y=x−3\(\begin{cases}\)y^2 = x^2 - 9 \\2y = x - 3\(\end{cases}\){y2=x2−92y=x−3

Verified video answer for a similar problem:

Key Concepts

Substitution Method

Solving Quadratic Equations

Checking Solutions in Systems

In Exercises 1–18, solve each system by the substitution method. $\begin{cases}\)y^2 = x^2 - 9 \\2y = x - 3\(\end{cases}$