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

문제 11

6장, 문제 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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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