In Exercises 47–48, solve each system by the method of your choice. (x + 2)/2 - (y + 4)/3 = 3 (x + y)/5 = (x - y)/2 - 5/2

검증된 단계별 안내

Start by rewriting each equation to eliminate the fractions for easier manipulation. For the first equation, multiply both sides by the least common multiple (LCM) of the denominators 2 and 3, which is 6, to clear the fractions.

For the first equation: multiply both sides by 6 to get: 6 * ((x + 2)/2 - (y + 4)/3) = 6 * 3. This simplifies to 3(x + 2) - 2(y + 4) = 18.

For the second equation, multiply both sides by the LCM of 5 and 2, which is 10, to clear the fractions: 10 * ((x + y)/5) = 10 * ((x - y)/2 - 5/2). This simplifies to 2(x + y) = 5(x - y) - 25.

Next, simplify both equations by distributing and combining like terms: For the first equation, expand 3(x + 2) and -2(y + 4). For the second equation, expand 2(x + y) and 5(x - y).

After simplification, you will have a system of two linear equations in standard form. Use either substitution or elimination method to solve for x and y.

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주요 개념

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

Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Solutions can be a single point, infinitely many points, or no solution, depending on the system's consistency.

Clearing Fractions in Equations

Clearing fractions involves multiplying both sides of an equation by the least common denominator to eliminate denominators. This simplifies the equation into a standard linear form, making it easier to manipulate and solve. It is especially useful when equations contain fractional expressions.

Methods for Solving Systems (Substitution, Elimination, or Graphing)

Common methods to solve systems include substitution (solving one equation for a variable and substituting into the other), elimination (adding or subtracting equations to eliminate a variable), and graphing (finding the intersection point of lines). Choosing the method depends on the system's form and complexity.