Problem 32 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the equation or the inequality. $$16+x^{2} \div 4=17 ; x=2$$

Step-by-Step Solution

Answer

Yes, \(x=2\) is a solution to the given equation.

Step 1: Substitute the Value of x into the Equation

Replace x in the equation with the given value i.e. 2. The equation becomes: \(16 + (2)^{2} \div 4 = 17\)

Step 2: Carry out the operations as per PEMDAS/BODMAS order

Begin by dealing with the exponent. After that:\[16 + (2)^{2} \div 4 = 17 => 16 + 4 \div 4 = 17\]Then proceed to division, yielding:\[16 + 4 \div 4 = 17 => 16 + 1 = 17\]

Step 3: Check if both sides of the equation are equal

Adding as per the equation results in 17, which is equal to the right side of the equation. Thus, \(x=2\) is a solution.

Key Concepts

Equation Solving

Equation solving is a fundamental aspect of algebra that involves finding the values of variables that make equations true. In our example, we are asked to determine if the given value, \(x = 2\), satisfies the equation \[ 16+x^{2} \div 4=17 \]. The goal is not just to plug in numbers but to methodically check if the equation holds true. Here's how it works:

• We substitute the value into the equation.
• Apply the rules of arithmetic (like PEMDAS/BODMAS) to simplify.
• Finally, verify if the equation is balanced, meaning both sides are equal once all operations are completed.

Understanding equation solving helps in breaking down complex algebraic expressions into simpler components, facilitating the process of verifying solutions.

PEMDAS/BODMAS

PEMDAS/BODMAS is a mnemonic that helps remember the order of operations in mathematics:

• Parentheses/Brackets
• Exponents/Orders (such as powers and square roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our example, this order is crucial. For the equation \( 16 + (2)^{2} \div 4 = 17 \), we first square the 2, giving us 4, and then divide by 4, yielding 1. Only after handling these operations do we add the 16, rendering the left side to match the right side of the equation as 17. It underscores the importance of carrying out operations in the correct sequence to ensure accuracy in solving equations.

Substitution Method

The substitution method is a way of solving equations, particularly when a given value for a variable is provided. It involves replacing the variable in the equation with the given value to see if the equation holds true.Here's how substitution works in our problem:

• Take the equation \( 16 + x^{2} \div 4 = 17 \).
• Substitute \(x = 2\) into the equation, making it \( 16 + (2)^{2} \div 4 = 17 \).
• Simplify the left side using the order of operations and check the result.

In this instance, upon simplifying, both sides of the equation equal 17, confirming that substituting \(x = 2\) makes the equation true. Thus, \(x = 2\) is indeed a solution to the equation, illustrating the substitution method’s power in equation solving.