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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 23
Chapter 6, Problem 23

Solve each system by elimination. In systems with fractions, first clear denominators.
5x + 7y = 6
10x - 3y = 46

Verified step by step guidance
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First, write down the system of equations clearly: \(5x + 7y = 6\) \(10x - 3y = 46\)
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. Notice that the coefficient of \(x\) in the second equation is \(10\), which is exactly twice the coefficient of \(x\) in the first equation (\(5\)).
Multiply the first equation by \(-2\) to make the coefficients of \(x\) opposites: \(-2(5x + 7y) = -2(6)\) which simplifies to \(-10x - 14y = -12\)
Now add this new equation to the second original equation: \((-10x - 14y) + (10x - 3y) = -12 + 46\) This will eliminate \(x\) and give you an equation with only \(y\).
Solve the resulting equation for \(y\). Once you find \(y\), substitute it back into one of the original equations to solve for \(x\).

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Key Concepts

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

System 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 other. This technique often requires multiplying one or both equations by constants to align coefficients before combining them.
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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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Radical Expressions with Fractions
Related Practice
Textbook Question

Solve each system by elimination. In systems with fractions, first clear denominators.

2x - 3y = -7

5x + 4y = 17

Textbook Question

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

Textbook Question

Find the inverse, if it exists, for each matrix.[1010−10211]\(\left\)[ \(\begin{matrix}\) 1 & 0 & 1 \\ 0 & -1 & 0 \\ 2 & 1 & 1 \(\end{matrix}\) \(\right\)]

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

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

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.

3x + 2y = -9

2x - 5y = -6

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