/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Solve each linear equation. $$... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 5

Solve each linear equation. $$ 15(y-9)=-60 $$

Short Answer

Expert verified
y = 5

Step by step solution

01

Distribute the 15

Multiply 15 by each term inside the parentheses: \[ 15(y-9) \to 15y - 135 \] So the equation becomes: \[ 15y - 135 = -60 \]
02

Isolate the term with y

Add 135 to both sides of the equation to isolate the term with y: \[ 15y - 135 + 135 = -60 + 135 \] Simplifying this gives: \[ 15y = 75 \]
03

Solve for y

Divide both sides by 15 to solve for y: \[ \frac{15y}{15} = \frac{75}{15} \] Simplifying this gives: \[ y = 5 \]

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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 an essential concept in algebra that allows us to remove parentheses. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. In mathematical terms, this is represented as: \[ a(b + c) = ab + ac \]
In our exercise, we have to distribute the 15 to both terms inside the parentheses (y-9). This step helps to simplify the equation and make it easier to solve. Here鈥檚 what that looks like:
\[ 15(y-9) \to 15y - 135 \]
Always remember, the distributive property works in both directions. You can use it to expand expressions and to factor them when needed.
isolating variables
Isolating variables is a crucial step in solving linear equations. The goal is to get the variable (in this case, y) alone on one side of the equation.
This process often involves using inverse operations to undo any addition, subtraction, multiplication, or division surrounding the variable.
In our example, after distributing the 15, we have:
\[ 15y - 135 = -60 \]
To isolate the y term, we add 135 to both sides of the equation to cancel out the -135. This gives us:
\[ 15y = 75 \]
Once the variable term is isolated, solving for the variable becomes straightforward.
basic algebra operations
Basic algebra operations include addition, subtraction, multiplication, and division. These operations are the building blocks for solving any algebraic equation.
Here鈥檚 how we applied them in our problem:
  • **Addition**: We added 135 to both sides to help isolate the variable.
  • **Multiplication**: We used the distributive property to multiply 15 by y and -9.
  • **Division**: Finally, we divided both sides by 15 to solve for y: \[ y = \frac{75}{15} = 5 \]

Knowing when and how to use these basic operations is key to solving any linear equation successfully. Mastery of these operations ensures you can handle more complex algebraic problems with ease.

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