Problem 11 - Prealgebra

Question

Find the solution of each equation from the list given. $$g+17=28 ; 9,11,13$$

Step-by-Step Solution

Answer

The solution is \( g = 11 \).

Step 1: Identify the Equation to Solve

We need to solve the equation \( g + 17 = 28 \). This is the equation given in the problem that we need to work with.

Step 2: Isolate the Variable

To find the value of \( g \), we should first isolate \( g \) on one side of the equation. Subtract 17 from both sides: \( g + 17 - 17 = 28 - 17 \)

Step 3: Simplify the Equation

After subtraction, the equation becomes:\( g = 11 \)Now \( g \) is isolated, and we've found its value.

Step 4: Check the Solution

Substitute \( g = 11 \) back into the original equation to ensure it's correct:\( 11 + 17 = 28 \)The equation holds true, confirming our solution is correct.

Step 5: Verify the Solution with Given Options

The options are 9, 11, and 13. Since we found \( g = 11 \), it matches one of the given options.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra where we find the value of an unknown variable that makes the equation true. The primary goal when solving an equation is to determine the value of this unknown variable.
Equations can be simple and linear, like the one in our exercise, or they can be more complex. In linear equations like \( g + 17 = 28 \), the variable is typically raised to the first power. Unlike quadratic or polynomial equations, linear equations have straightforward solutions. Here’s how you generally solve them:

• Identify the equation: Look at what is given in the equation. In this case, it is \( g + 17 = 28 \).
• Simplify if necessary: This might involve combining like terms or removing parentheses.
• Isolate the variable: Use operations like addition, subtraction, multiplication, or division to get the variable alone on one side.
• Check your solution: Substitute your found variable back into the original equation to verify your solution.

Mastering these steps allows you to solve basic algebraic equations with ease.

Prealgebra

Prealgebra forms the mathematical foundation necessary before moving on to more advanced algebraic concepts. It often includes understanding basic arithmetic and learning to manipulate simple equations and expressions.
In prealgebra, you'll encounter terms like variables, constants, coefficients, and expressions. These are crucial for understanding and setting up equations. Let's break it down:

• Variables: Symbols used to represent unknown values, such as \( g \) in our exercise.
• Constants: Fixed values within equations, like the numbers 17 and 28 in the given equation.
• Coefficients: Numbers that multiply the variable. In \( 2x \), the number 2 is a coefficient.
• Expressions: Combinations of variables, numbers, and operations (like addition or subtraction) that make up parts of an equation.

Understanding these concepts in prealgebra sets you up for solving real-world mathematical problems.

Variable Isolation

Variable isolation is a technique in algebra used to solve for an unknown variable by getting it alone on one side of the equation. It makes solving equations systematic and straightforward.
The goal is to perform operations on both sides of an equation to "move" everything but the variable to the other side. Here's how we do it:

• Look at the equation: Let's use \( g + 17 = 28 \) as an example. Our goal is to solve for \( g \).
• Perform the opposite operation: Since \( g \) is added to 17, do the opposite subtraction to both sides: \( g + 17 - 17 = 28 - 17 \).
• Simplify both sides: This leaves \( g = 11 \), effectively isolating the variable.

Variable isolation not only helps solve equations but is an essential skill across all mathematical disciplines, providing clarity and accuracy in problem-solving.