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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 1
Chapter 6, Problem 1

Solve each system by the substitution method.
{x+y=2y=x2−4\(\begin{cases}\)x + y = 2 \(\y\) = x^2 - 4\(\end{cases}\)

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1
Start with the given system of equations: \(x + y = 2\) and \(y = x^{2} - 4\).
From the first equation, solve for \(y\) in terms of \(x\): \(y = 2 - x\).
Substitute this expression for \(y\) into the second equation: \(2 - x = x^{2} - 4\).
Rewrite the equation to set it equal to zero: \(x^{2} - 4 - (2 - x) = 0\), which simplifies to \(x^{2} - 4 - 2 + x = 0\).
Combine like terms to get a quadratic equation: \(x^{2} + x - 6 = 0\). This quadratic can now be solved using factoring, completing the square, or the quadratic formula.

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Key Concepts

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

System of Equations

A system of equations consists of two or more equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. In this problem, the system includes a linear and a quadratic equation involving x and y.
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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. Here, y is expressed from the first equation and substituted into the second.
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Solving Quadratic Equations

After substitution, the resulting equation is quadratic, which can be solved by factoring, completing the square, or using the quadratic formula. The solutions to the quadratic give possible values for x, which can then be used to find corresponding y values.
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Related Practice
Textbook Question

Determine whether the given ordered pair is a solution of the system. (2,3)(2, 3)

{x+3y=11x−5y=−13\(\begin{cases}\)x + 3y = 11 \(\x\) - 5y = -13\(\end{cases}\)

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Textbook Question

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

{y=4x+13x+2y=13\(\begin{cases}\) y = 4x + 1 \\ 3x + 2y = 13 \(\end{cases}\)

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Textbook Question

Find the value of the objective function at each corner of the graphed region. What is the maximum value of the objective function? What is the minimum value of the objective function? 1. Objective Function z=5x+6y


Textbook Question

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.(11x - 10)/(x − 2) (x + 1)

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Textbook Question

Determine if the given ordered triple is a solution of the system. (2,−1,3)(2,−1, 3)

{x+y+z=4x−2y−z=12x−y−2z=−1\(\begin{cases}\)x + y + z = 4 \(\x\) - 2y - z = 1 \\2x - y - 2z = -1\(\end{cases}\)

Textbook Question

Ggraph each inequality. x+2y≤8

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