Evaluate x2 - (xy - y) for x satisfying 3(x + 3)/5 = 2x + 6 and y satisfying - 2y - 10 = 5y + 18.

Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 97a

2장, 문제 97a

Evaluate x² - (xy - y) for x satisfying 3(x + 3)/5 = 2x + 6 and y satisfying - 2y - 10 = 5y + 18.

검증된 단계별 안내

Step 1: Solve for x in the equation \( \frac{3(x + 3)}{5} = 2x + 6 \). Start by eliminating the fraction by multiplying through by 5, resulting in \( 3(x + 3) = 5(2x + 6) \). Expand both sides to get \( 3x + 9 = 10x + 30 \).

Step 2: Rearrange the equation \( 3x + 9 = 10x + 30 \) to isolate x. Subtract \( 3x \) from both sides to get \( 9 = 7x + 30 \). Then subtract 30 from both sides to get \( -21 = 7x \). Finally, divide both sides by 7 to solve for x.

Step 3: Solve for y in the equation \( -2y - 10 = 5y + 18 \). Start by isolating y by adding \( 2y \) to both sides, resulting in \( -10 = 7y + 18 \). Subtract 18 from both sides to get \( -28 = 7y \). Finally, divide both sides by 7 to solve for y.

Step 4: Substitute the values of x and y into the expression \( x^2 - (xy - y) \). Start by calculating \( x^2 \), then calculate \( xy \), and finally calculate \( xy - y \).

Step 5: Subtract \( (xy - y) \) from \( x^2 \) to evaluate the expression. Simplify the result to complete the problem.

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주요 개념

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

Solving Linear Equations

To evaluate the expression, we first need to solve the linear equations for x and y. A linear equation is an equation of the first degree, meaning it can be written in the form ax + b = 0. We can isolate the variable by performing algebraic operations, such as addition, subtraction, multiplication, or division, to find the values of x and y that satisfy the equations.

Substitution

Once we have the values of x and y, we can substitute these values into the expression x^2 - (xy - y). Substitution is a fundamental algebraic technique where we replace a variable with its corresponding value. This allows us to simplify the expression and compute the final result based on the values obtained from the linear equations.

Evaluating Expressions

Evaluating an expression involves calculating its value after substituting the variables with their respective numerical values. In this case, we will compute x^2 - (xy - y) using the values of x and y found earlier. Understanding how to correctly perform operations such as addition, subtraction, multiplication, and exponentiation is crucial for obtaining the correct final answer.