Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 39

6장, 문제 39

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. x/4 - y/4 = −1 x + 4y = -9

검증된 단계별 안내

Start by writing the system of equations clearly:

\frac{x}{4} - \frac{y}{4} = -1

and

x + 4y = -9

Multiply the first equation by 4 to eliminate the denominators:

4 \(\times\) \(\left\)( \(\frac{x}{4}\) - \(\frac{y}{4}\) \(\right\)) = 4 \(\times\) (-1)

, which simplifies to

x - y = -4

Now you have the system:

x - y = -4

and

x + 4y = -9

. Use either substitution or elimination to solve this system. For elimination, subtract the first equation from the second to eliminate

x

(x + 4y) - (x - y) = -9 - (-4)

Simplify the subtraction:

x + 4y - x + y = -9 + 4

, which reduces to

5y = -5

. Solve for

y

by dividing both sides by 5.

Substitute the value of

y

back into one of the original equations (for example,

x - y = -4

) to solve for

x

. After finding both

x

and

y

, express the solution as an ordered pair

(x, y)

. If you find contradictory statements or infinite solutions during the process, identify and express those accordingly using set notation.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Solving Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding all variable values that satisfy all equations simultaneously. Common methods include substitution, elimination, and graphing.

Types of Solutions for Systems

Systems of linear equations can have one solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (dependent). Identifying the type depends on the relationships between the equations, such as parallel lines or coincident lines.

Expressing Solution Sets Using Set Notation

Set notation is a concise way to represent all solutions of a system. For example, a single solution is written as {(x, y)}, no solution as the empty set ∅, and infinitely many solutions as a set describing the relationship between variables, like {(x, y) | y = 2x + 3}.