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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 49
Chapter 6, Problem 49

In Exercises 47–52, solve each system by the method of your choice. {−4x+y=12y=x3+3x2\(\begin{cases}\)-4x + y = 12 \(\y\) = x^3 + 3x^2\(\end{cases}\)

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Start with the given system of equations: \(-4x + y = 12\) and \(y = x^3 + 3x^2\).
Since both equations are equal to \(y\), set the right-hand sides equal to each other to eliminate \(y\): \(-4x + y = 12\) implies \(y = 4x + 12\) (by adding \$4x$ to both sides), so set \(4x + 12 = x^3 + 3x^2\).
Rewrite the equation to bring all terms to one side, forming a polynomial equation: \(x^3 + 3x^2 - 4x - 12 = 0\).
Solve the cubic equation \(x^3 + 3x^2 - 4x - 12 = 0\) by factoring or using methods such as the Rational Root Theorem to find possible roots for \(x\).
Once you find the values of \(x\), substitute each back into one of the original equations (for example, \(y = 4x + 12\)) to find the corresponding \(y\) values, giving you the solution pairs \((x, y)\).

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Key Concepts

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

Systems of Equations

A system of equations consists of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Solutions can be points where the graphs of the equations intersect.
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Introduction to Systems of Linear Equations

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.
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Choosing a Method to Solve Quadratics

Solving Nonlinear Equations

When a system includes nonlinear equations, such as cubic or quadratic terms, solving requires careful algebraic manipulation. After substitution, the resulting equation may be polynomial, requiring factoring or other methods to find roots.
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Related Practice
Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

{(x−1)2+(y+1)2<25(x−1)2+(y+1)2≥16\(\begin{cases}\)(x - 1)^2 + (y + 1)^2 < 25 \\(x - 1)^2 + (y + 1)^2 \(\geq\) 16\(\end{cases}\)

Textbook Question

In Exercises 49–50, solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0, b ≠ 0 5ax + 4y = 17 ax + 7y = 22

Textbook Question

In Exercises 47–52, solve each system by the method of your choice. {3x2+1y2=75x2−2y2=−3\(\begin{cases}\]\frac{3}{x^2}\) + \(\frac{1}{y^2}\) = 7 \(\frac{5}{x^2}\) - \(\frac{2}{y^2}\) = -3\(\end{cases}\)

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Textbook Question

Find the partial fraction decomposition for 1/x(x+1) and use the result to find the following sum:

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Textbook Question

In Exercises 47–48, solve each system by the method of your choice. (x - y)/3 = (x + y)/2 - 1/2 (x + 2)/2 - 4 = (y + 4)/3

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Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

{x2+y2>1x2+y2<16\(\begin{cases}\)x^2 + y^2 > 1 \(\x\)^2 + y^2 < 16\(\end{cases}\)