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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 21
Chapter 6, Problem 21

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

Verified step by step guidance
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

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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Textbook Question

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

703
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Textbook Question

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

662
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Textbook Question

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 \(\end{matrix}\) \(\right\)]

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Textbook Question

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

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Textbook Question

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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