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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 1
Chapter 6, Problem 1

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,____).
x + y = 7
x2 + y2 = 25

Verified step by step guidance
1
Start with the given system of equations: \(x + y = 7\) and \(x^2 + y^2 = 25\).
Since one solution has \(x = 3\), substitute \(x = 3\) into the first equation to find \(y\): \(3 + y = 7\).
Solve for \(y\) in the equation \(3 + y = 7\) to find the value of \(y\) corresponding to \(x = 3\).
Verify the solution \((3, y)\) by substituting both \(x = 3\) and the found \(y\) into the second equation \(x^2 + y^2 = 25\) to ensure it satisfies the equation.
To find the other solution, use the substitution \(y = 7 - x\) from the first equation and substitute into the second equation to form a quadratic in \(x\), then solve for \(x\) and \(y\).

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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

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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Substitution Method

The substitution method involves solving one equation for one variable and then 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 is nonlinear.
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Interpreting Solutions of Nonlinear Systems

Nonlinear systems can have multiple solutions corresponding to the intersection points of curves like circles and lines. Each solution is an ordered pair (x, y) that satisfies all equations. Understanding the geometric meaning helps verify solutions and interpret the number and nature of solutions.
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Related Practice
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What is the value of ∣40−20∣\(\left\)| \(\begin{matrix}\) 4 & 0 \\ -2 & 0 \(\end{matrix}\) \(\right\)|?

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Answer each question. What is the product of [64−18]\(\left\)[ \(\begin{matrix}\) 6 & 4 \\ -1 & 8 \(\end{matrix}\) \(\right\)] and I2 (in either order)?

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Answer each question. What is the product [−574230−166][100010001]\(\left\)[ \(\begin{matrix}\) -5 & 7 & 4 \\2 & 3 & 0 \\-1 & 6 & 6 \(\end{matrix}\) \(\right\)]\(\quad\]\left\)[ \(\begin{matrix}\) 1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \(\end{matrix}\) \(\right\)]?

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Textbook Question

Fill in the blank(s) to correctly complete each sentence. For the following statement to be true, the value of x must be ____, and the value of y must be ____.

[3−651]=[x+1−65y+1]\(\left\)[ \(\begin{matrix}\) 3 & -6 \\ 5 & 1 \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) x + 1 & -6 \\ 5 & y + 1 \(\end{matrix}\) \(\right\)]

Textbook Question

Match each system of inequalities with the correct graph from choices A–D. x ≥ 5 y ≤ -3