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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 5
Chapter 6, Problem 5

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. If we want to solve the following nonlinear system by substitution and we decide to solve equation (2) for y, what will be the resulting equation when the substitution is made into equation (1)?
x2 + y = 2 (1)
x - y = 0 (2)

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1
Identify the system of equations: (1) \(x^2 + y = 2\) and (2) \(x - y = 0\).
Solve equation (2) for \(y\): since \(x - y = 0\), rearrange to get \(y = x\).
Substitute the expression for \(y\) from equation (2) into equation (1): replace \(y\) in \(x^2 + y = 2\) with \(x\).
After substitution, the resulting equation becomes \(x^2 + x = 2\).
This new equation in terms of \(x\) can now be solved to find the values of \(x\), which can then be used to find \(y\).

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Key Concepts

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

Solving Systems of Equations by Substitution

Substitution involves solving one equation for one variable and then replacing that variable in the other equation. This method reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one equation is easily solved for one variable.
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Isolating a Variable

Isolating a variable means rewriting an equation so that one variable is expressed explicitly in terms of the others. For example, solving equation (2) for y means rewriting it as y = x. This step is crucial for substitution because it provides the expression to replace the variable in the other equation.
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Equations with Two Variables

Substituting Expressions into Equations

After isolating a variable, substitution involves replacing that variable in the other equation with the expression found. For instance, substituting y = x into equation (1) transforms it into an equation with only x, such as x^2 + x = 2, which can then be solved using algebraic methods.
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Related Practice
Textbook Question

What is the value of x if ∣x00x∣=9\(\begin{vmatrix}\)x & 0\\ 0 & x\(\end{vmatrix}\)=9?

Textbook Question

Answer each question. By what expression should we multiply each side of (3x - 1)/(x(2x^2 + 1)^2) = A/x + (Bx + C)/(2x^2 + 1) + (Dx + E)/(2x^2 + 1)^2 so that there are no fractions in the equation?

Textbook Question

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.

16x+13y=8\(\frac{1}{6x}\)+\(\frac{1}{3y}\)=8

14x+12y=12\(\frac{1}{4x}\)+\(\frac{1}{2y}\)=12

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Textbook Question

Evaluate each determinant.

∣−594−1∣\(\left\)| \(\begin{matrix}\) -5 & 9 \\ 4 & -1 \(\end{matrix}\) \(\right\)|

Textbook Question

If the dimension of matrix A is 3 ×\(\times\) 2 and the dimension of matrix B is 2 ×\(\times\) 6, then the dimension of AB is ____.

Textbook Question

Find the dimension of each matrix. Identify any square, column, or row matrices.

[−4823]\(\left\)[ \(\begin{matrix}\) -4 & 8 \\ 2 & 3 \(\end{matrix}\) \(\right\)]