Textbook Question
Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle.
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 7a
Solve and check each linear equation. 11x - (6x - 5) = 40
Verified step by step guidance1
Distribute the negative sign across the parentheses in the equation. This means rewriting \( 11x - (6x - 5) \) as \( 11x - 6x + 5 \). The equation becomes \( 11x - 6x + 5 = 40 \).
Combine like terms on the left-hand side of the equation. Combine \( 11x \) and \( -6x \) to get \( 5x \). The equation simplifies to \( 5x + 5 = 40 \).
Isolate the variable term \( 5x \) by subtracting 5 from both sides of the equation. This gives \( 5x = 35 \).
Solve for \( x \) by dividing both sides of the equation by 5. This gives \( x = \frac{35}{5} \).
Check your solution by substituting \( x \) back into the original equation \( 11x - (6x - 5) = 40 \) and verifying that both sides are equal.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Equations
A linear equation is an algebraic expression that represents a straight line when graphed. It typically takes the form ax + b = c, where a, b, and c are constants, and x is the variable. Solving a linear equation involves isolating the variable on one side of the equation to find its value.
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Categorizing Linear Equations
Distributive Property
The distributive property is a fundamental algebraic principle that states a(b + c) = ab + ac. This property allows us to eliminate parentheses in expressions by distributing the multiplier across the terms inside the parentheses. In the context of the given equation, it helps simplify expressions like 6x - 5.
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Checking Solutions
Checking solutions involves substituting the found value of the variable back into the original equation to verify its correctness. This step ensures that the solution satisfies the equation, confirming that no errors were made during the solving process. It is a crucial part of solving linear equations.
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Related Practice
Textbook Question
A new car worth \$45,000 is depreciating in value by \$5000 per year. a. Write a formula that models the car's value, y, in dollars, after x years. b. Use the formula from part (a) to determine after how many years the car's value will be \$10,000. c. Graph the formula from part (a) in the first quadrant of a rectangular coordinate system. Then show your solution to part (b) on the graph.
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Textbook Question
Solve each equation in Exercises 1 - 14 by factoring.
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Textbook Question
Solve and check each linear equation. x - 5(x + 3) = 13
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Textbook Question
You are choosing between two gyms. One gym offers membership for a fee of \$40 plus a monthly fee of \$25. The other offers membership for a fee of \$15 plus a monthly fee of \$30. After how many months will the total cost at each gym be the same? What will be the total cost for each gym?
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Textbook Question
In Exercises 1–8, add or subtract as indicated and write the result in standard form. 8i - (14 - 9i)
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