교과서 질문
Solve each system by substitution.
4x + 3y = -13
-x + y = 5
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Ch. 5 - Systems and Matrices
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
6장, 문제 8
Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary.
6x + 10y = -11
9x + 6y = -3
검증된 단계별 안내1
First, write down the system of equations clearly:
\[6x + 10y = -11\]
\[9x + 6y = -3\]
Choose a method to solve the system. Here, we will use the elimination method. The goal is to eliminate one variable by making the coefficients of either \(x\) or \(y\) the same (or opposites) in both equations.
To eliminate \(x\), find the least common multiple (LCM) of the coefficients of \(x\) in both equations, which are 6 and 9. The LCM is 18. Multiply the first equation by 3 and the second equation by -2 to get:
\[3(6x + 10y) = 3(-11) \Rightarrow 18x + 30y = -33\]
\[-2(9x + 6y) = -2(-3) \Rightarrow -18x - 12y = 6\]
Add the two new equations to eliminate \(x\):
\[(18x + 30y) + (-18x - 12y) = -33 + 6\]
This simplifies to:
\[18x - 18x + 30y - 12y = -27\]
\[0x + 18y = -27\]
So, you get:
\[18y = -27\]
Solve for \(y\) by dividing both sides by 18:
\[y = \frac{-27}{18}\]
Once you have \(y\), substitute this value back into one of the original equations to solve for \(x\). This will give you the solution to the system. If you find a contradiction or a true statement for all values, you can identify if the system is inconsistent or has infinitely many solutions.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Substitution and Elimination Methods
These are algebraic techniques used to solve systems of linear equations. Substitution involves solving one equation for a variable and substituting it into the other, while elimination involves adding or subtracting equations to eliminate a variable, simplifying the system to one equation with one variable.
추천 영상:
가이드 코스
6:04
How to Multiply Equations in Elimination Method
Types of Solutions for Systems of Equations
A system can have a unique solution (one point), infinitely many solutions (all points on a line), or no solution (parallel lines). Identifying the type depends on the relationships between the equations, such as whether they are multiples of each other or contradictory.
추천 영상:
가이드 코스
4:27Introduction to Systems of Linear Equations
Expressing Solutions with Parameters
When a system has infinitely many solutions, the solution set is expressed using a parameter (like y arbitrary) to represent all possible solutions. This involves rewriting one variable in terms of the other, showing the dependency between variables.
추천 영상:
가이드 코스
05:09Introduction to Algebraic Expressions
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