Problem 83 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is \(-4\) a solution of \(x-9=5 ?\)

Step-by-Step Solution

Answer

-4 is not a solution, because -13 does not equal 5.

Step 1: Substitute the Given Number

Replace the variable \(x\) in the equation \(x - 9 = 5\) with the given number, \(-4\). This gives us the new equation: \(-4 - 9 = 5\).

Step 2: Perform the Arithmetic

Calculate the left side of the equation: \(-4 - 9 = -13\).

Step 3: Compare Both Sides of the Equation

Check if the calculated left side, \(-13\), is equal to the right side, \(5\). Since \(-13\) is not equal to \(5\), \(-4\) is not a solution of the equation.

Key Concepts

Substitution Method

The substitution method is a crucial part of algebra that involves replacing a variable with a given number to test if it satisfies an equation. In this exercise, we are checking if

• the number \(-4\) is a solution to the equation.
• To do this, we substitute \(-4\) for \(x\) in the equation \(x - 9 = 5\).
• This gives us a new equation: \(-4 - 9 = 5\).

Substitution helps determine whether the specific value makes the equation true. It is a method widely used in algebra to solve equations or verify potential solutions. It's like testing a key to see if it fits in a lock — if the key doesn't fit, then it's not the right solution!

Arithmetic Operations

Arithmetic operations are basic mathematical procedures including addition, subtraction, multiplication, and division. In this particular problem, we use subtraction, one of the simplest forms of arithmetic.

• We substitute the value \(-4\) into the equation, we subtract \(9\) from \(-4\), giving us: \(-4 - 9\).
• This operation results in \(-13\),

Understanding how to properly execute these operations is fundamental for solving and simplifying equations. Missing out on accuracy with arithmetic can lead to errors in determining whether a number is a solution or not. Always pay attention to the signs (positive or negative) during subtraction to ensure correctness.

Checking Equality

Checking for equality in an equation is key to verifying a solution. The main idea is to ensure that both sides of the equation have the same value after substitution and arithmetic operations.

• We computed the left side: \(-13\), and we were given the right side \(5\).
• We then compared these two results: \(-13\) and \(5\).

For \(-4\) to be a solution, these values need to match. But since \(-13\) does not equal \(5\), it indicates that \(-4\) is not a solution to the equation. Always ensure the balance of both sides for equality when testing solutions.

Beginning Algebra

Beginning algebra focuses on understanding and solving equations. This involves learning concepts such as variables, expressions, and basic operations. The goal is to find the unknowns, or verify potential solutions like in our problem.

• We had a basic linear equation \(x - 9 = 5\) to solve.
• The term \(x\) is our unknown, a common feature in algebra.

Algebra equips students with problem-solving strategies that are useful in more complex mathematics and everyday life. Establishing a solid foundation in beginning algebra paves the way for tackling more challenging algebraic concepts confidently. By practicing substitution and equality checking, students enhance their skills and become more proficient in solving equations.