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Chapter 8: Problem 162

Solve the linear equation using the general strategy. $$5(x+3)=75$$

Short Answer

Expert verified
x = 12

Step by step solution

01

Distribute the 5

Apply the distributive property to remove the parentheses by multiplying 5 by each term inside the parentheses: 5(x + 3) = 75 This becomes: 5x + 15 = 75
02

Isolate the variable term

Subtract 15 from both sides to isolate the term with the variable (5x): 5x + 15 - 15 = 75 - 15 This simplifies to: 5x = 60
03

Solve for x

Divide both sides by 5 to solve for x: \[ x = \frac{60}{5} \] This simplifies to: x = 12

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra. It allows us to simplify expressions by distributing a multiplication operation over addition or subtraction inside parentheses. The formula for the distributive property is: \[ a(b + c) = ab + ac \]Starting with the equation:\[ 5(x + 3) = 75 \] we apply the distributive property: \[ 5 \times x + 5 \times 3 = 75 \] This results in:\[ 5x + 15 = 75 \] Using the distributive property allows us to eliminate parentheses, making the equation simpler and easier to solve for the variable.
Isolating the Variable
Isolating the variable is a critical skill in solving linear equations. The goal is to get the variable by itself on one side of the equation, making it easier to solve. In the equation:\[ 5x + 15 = 75 \] we need to isolate the term containing the variable, 5x. To do this, subtract 15 from both sides: \[ 5x + 15 - 15 = 75 - 15 \] This simplifies to: \[ 5x = 60 \]By removing the constant (15), the variable term is isolated, and the equation is simpler: we can now solve directly for x.
Basic Algebra
Basic algebra involves using various techniques to solve equations. One of these techniques includes isolating the variable, as shown in the previous section. Let's continue solving the equation:\[ 5x = 60 \] To solve for x, divide both sides of the equation by the coefficient of x (in this case, 5): \[ x = \frac{60}{5} \] This simplifies to: \[ x = 12 \]Basic algebra focuses on operations such as addition, subtraction, multiplication, and division to manipulate and solve equations. Mastering these basic techniques allows you to solve a wide range of mathematical problems.

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Most popular questions from this chapter

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