Solve each system. (Hint: In Exercises 69–72, let 1/x=t1/x = t and 1/y=u1/y = u.)2x+3y=18\frac{2}{x}+\frac{3}{y}=184x−5y=−8\frac{4}{x}-\frac{5}{y}=-8

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 72

6장, 문제 72

Solve each system. (Hint: In Exercises 69–72, let $1 / x = t$ and $1 / y = u$ .)
$\frac{2}{x} + \frac{3}{y} = 18$
$\frac{4}{x} - \frac{5}{y} = - 8$

검증된 단계별 안내

Start by using the hint given: let $ t = \frac{1}{x} $ and $ u = \frac{1}{y} $. This substitution will transform the system into a linear system in terms of $ t $ and $ u $.

Rewrite each equation by substituting $ \frac{1}{x} = t $ and $ \frac{1}{y} = u $. The system becomes: \[ 2t + 3u = 18 \] \[ 4t - 5u = -8 \]

Solve the new system of linear equations for $ t $ and $ u $ using either the substitution method or the elimination method. For example, you can multiply the first equation by 4 and the second by 2 to align coefficients for elimination.

Once you find the values of $ t $ and $ u $, recall that $ t = \frac{1}{x} $ and $ u = \frac{1}{y} $. Use these relationships to solve for $ x $ and $ y $ by taking the reciprocal of $ t $ and $ u $ respectively.

Check your solutions by substituting $ x $ and $ y $ back into the original equations to ensure they satisfy both equations.

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영상 길이:

주요 개념

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

Substitution Method

The substitution method involves replacing variables with new expressions to simplify a system of equations. In this problem, substituting 1/x = t and 1/y = u transforms the original system into a linear system in terms of t and u, making it easier to solve.

Solving Systems of Linear Equations

Once the substitution is made, the system becomes linear in t and u. Solving systems of linear equations involves finding values for variables that satisfy all equations simultaneously, typically using methods like substitution, elimination, or matrix operations.

Back-Substitution

After finding the values of t and u, back-substitution is used to find the original variables x and y by reversing the substitution: x = 1/t and y = 1/u. This step is crucial to interpret the solution in terms of the original variables.

Solve each system. (Hint: In Exercises 69–72, let 1/x=t1/x = t and 1/y=u1/y = u.)2x+3y=18\(\frac{2}{x}\)+\(\frac{3}{y}\)=184x−5y=−8\(\frac{4}{x}\)-\(\frac{5}{y}\)=-8