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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 15
Chapter 6, Problem 15

In Exercises 1–18, solve each system by the substitution method. {x+y=1x2+xy−y2=−5\(\begin{cases}\)x + y = 1 \(\x\)^2 + xy - y^2 = -5\(\end{cases}\)

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1
Start with the given system of equations: \(x + y = 1\) and \(x^2 + xy - y^2 = -5\).
From the first equation, solve for one variable in terms of the other. For example, solve for \(x\): \(x = 1 - y\).
Substitute the expression for \(x\) from step 2 into the second equation: \((1 - y)^2 + (1 - y) y - y^2 = -5\).
Expand and simplify the substituted equation by applying algebraic operations such as expanding squares and distributing terms: Expand \((1 - y)^2\) to \(1 - 2y + y^2\), then combine like terms carefully.
After simplification, you will get a quadratic equation in terms of \(y\). Solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the possible values of \(y\). Then, substitute these values back into \(x = 1 - y\) to find the corresponding \(x\) values.

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Key Concepts

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

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 easily solved for one variable.
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Solving Nonlinear Systems

Nonlinear systems include equations where variables are raised to powers or multiplied together, such as quadratic terms. Solving these requires careful algebraic manipulation and may yield multiple solutions. Understanding how to handle nonlinear terms is essential for finding all valid solutions.
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Quadratic Equations

Quadratic equations are polynomial equations of degree two, often in the form ax^2 + bx + c = 0. They can be solved by factoring, completing the square, or using the quadratic formula. Recognizing and solving quadratic equations is key when substitution leads to a quadratic expression.
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Related Practice
Textbook Question

In Exercises 16–24, write the partial fraction decomposition of each rational expression. x/(x - 3)(x + 2)

Textbook Question

Find the quadratic function y = ax^2 + bx + c whose graph passes through the points (1, 4), (3, 20), and (-2, 25).

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Textbook Question

In Exercises 5–18, solve each system by the substitution method. 2x - 3y = 8 - 2x 3x + 4y = x + 3y + 14

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Textbook Question

Solve each system in Exercises 5–18. {x+y=−4y−z=12x+y+3z=−21\(\begin{cases}\)x + y = -4 \(\y\) - z = 1 \\2x + y + 3z = -21\(\end{cases}\)

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Textbook Question

Write the partial fraction decomposition of each rational expression. 4/(2x2 -5x -3)

Textbook Question

In Exercises 1–26, graph each inequality. x2+y2>25

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