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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Lial 13th EditionCh. 5 - Systems and Matrices문제 21
6장, 문제 21

Solve each system by elimination. In systems with fractions, first clear denominators.
2x - 3y = -7
5x + 4y = 17

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1
Identify the system of equations: \[2x - 3y = -7\] \[5x + 4y = 17\]
To use the elimination method, we want to eliminate one variable by making the coefficients of either \(x\) or \(y\) the same (or opposites) in both equations. Let's choose to eliminate \(y\).
Find the least common multiple (LCM) of the coefficients of \(y\), which are 3 and 4. The LCM is 12. Multiply the first equation by 4 and the second equation by 3 to get matching coefficients for \(y\): \[4(2x - 3y) = 4(-7) \Rightarrow 8x - 12y = -28\] \[3(5x + 4y) = 3(17) \Rightarrow 15x + 12y = 51\]
Add the two new equations to eliminate \(y\): \[(8x - 12y) + (15x + 12y) = -28 + 51\] This simplifies to: \[8x + 15x = 23x\] \[-12y + 12y = 0\] \[-28 + 51 = 23\] So, the resulting equation is: \[23x = 23\]
Solve for \(x\) by dividing both sides by 23: \[x = \frac{23}{23}\] Once you find \(x\), substitute it back into one of the original equations to solve for \(y\).

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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 with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and set up these systems is essential for solving them.
추천 영상:
가이드 코스
4:27
Introduction to Systems of Linear Equations

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable. This technique often requires multiplying one or both equations by constants to align coefficients before elimination.
추천 영상:
가이드 코스
6:04
How to Multiply Equations in Elimination Method

Clearing Fractions

When equations contain fractions, multiplying both sides by the least common denominator removes the fractions, simplifying calculations. Clearing denominators helps avoid errors and makes the elimination process more straightforward.
추천 영상:
가이드 코스
05:45
Radical Expressions with Fractions
관련 실천
교과서 질문

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

x + y = 5

x - y = -1

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교과서 질문

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-3)/(x2(x2 + 5))

703
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교과서 질문

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x3 + 4)/(9x3 - 4x)

662
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교과서 질문

Find the values of the variables for which each statement is true, if possible.

[−7+z4r8s6p25]+[−98r3254]=[2362720712a]\(\left\)[ \(\begin{matrix}\) -7 + z & 4r & 8s \\ 6p & 2 & 5 \(\end{matrix}\) \(\right\)] + \(\left\)[ \(\begin{matrix}\) -9 & 8r & 3 \\ 2 & 5 & 4 \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) 2 & 36 & 27 \\ 20 & 7 & 12a \(\end{matrix}\) \(\right\)]

82
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교과서 질문

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

x2+y2=5x^2+y^2=5

−3x+4y=2-3x+4y=2

497
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교과서 질문

Find the inverse, if it exists, for each matrix. [510−3−6]\(\left\)[ \(\begin{matrix}\) 5 & 10 \\ -3 & -6 \(\end{matrix}\) \(\right\)]

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