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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 3
Chapter 6, Problem 3

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (___, 3).
2x + y = 1
x2 + y2 = 10

Verified step by step guidance
1
Start by substituting the given y-value into the first equation. Since one solution has y = 3, plug y = 3 into the linear equation \(2x + y = 1\) to find the corresponding x-value.
Rewrite the first equation with y = 3: \(2x + 3 = 1\). Then, solve for \(x\) by isolating it on one side: \(2x = 1 - 3\).
Simplify the right side to get \(2x = -2\), and then divide both sides by 2 to find \(x = -1\).
Now, verify this solution by substituting \(x = -1\) and \(y = 3\) into the second equation \(x^2 + y^2 = 10\) to ensure it satisfies the equation.
To find the other solution, use the system of equations: express \(y\) from the first equation as \(y = 1 - 2x\), then substitute this expression into the second equation \(x^2 + y^2 = 10\) and solve the resulting quadratic equation for \(x\). Finally, find the corresponding \(y\) values.

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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

A system of equations consists of two or more equations with the same variables. Solving the system means finding all variable values that satisfy every equation simultaneously. Methods include substitution, elimination, and graphing, which help find points where the equations intersect.
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Solving Systems of Equations - Substitution

Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. It is especially useful when one equation is linear and the other nonlinear.
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Nonlinear Equations and Circles

Nonlinear equations, such as circles defined by x² + y² = r², represent curves rather than lines. Understanding the geometric meaning helps interpret solutions as intersection points between a line and a curve. These intersections correspond to the system's solutions.
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Related Practice
Textbook Question

What expression in x represents ∣x43x∣\(\left\)| \(\begin{matrix}\) x & 4 \\ 3 & x \(\end{matrix}\) \(\right\)|?

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Textbook Question

Answer each question. By what expression should we multiply each side of (3x - 2)/(x + 4)(3x^2 + 1) = A/(x + 4) + (Bx + C)/(3x^2 + 1) so that there are no fractions in the equation?

Textbook Question

What is the augmented matrix of the following system?

-3x + 5y = 2

6x + 2y = 7

6
views
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

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 ____.