Problem 14 - Algebra 1: Concepts and Skills

Question

Check to see if \(a=5\) is or is not a solution of the equation. $$ 45 \div a=9 $$

Step-by-Step Solution

Answer

Yes, \(a=5\) is a solution to the equation.

Step 1: Identify the Equation and the Proposed Solution

The equation that needs to be checked is \(45 \div a = 9\), and the proposed solution is \(a=5\)

Step 2: Substitute the Value of 'a' in the Equation

Replacing variable 'a' in the equation with 5 results in this equation: \(45 \div 5 = 9\)

Step 3: Perform the Calculation

The left side of the equation after substitution is \(45 \div 5\) which equals 9

Step 4: Verify Solution

As the two sides of the equation are equal, it verifies that \(a=5\) is indeed a solution for the given equation.

Key Concepts

Understanding Division

Division is like the opposite of multiplication. It involves splitting a number into several equal parts. When you divide, you are essentially asking: "How many times does one number fit into another?" For example, let's look at the division problem in our equation:

• The equation is \(45 \div a = 9\).
• This means we want to know how many times the number \(a\) fits into 45 to get 9.

To calculate this, you can think of it as finding some number multiplied by 9 to reach 45. Here, when \(a\) is 5,

• \(45 \div 5 = 9\) because 5 multiplied by 9 gives us 45.

This is how division helps us rearrange and solve equations by looking at one number as a factor of another.

Concept of Variable Substitution

Variable substitution means replacing a variable with a specific value in an equation. Variables, like 'a' in our equation, stand in for unknown numbers. By substituting, we test if a particular value makes the equation true.

• First, identify the equation and the variable you want to replace, which in this case is \(45 \div a = 9\) and \(a = 5\).
• Next, replace \(a\) with 5: \(45 \div 5 = 9\).
• After substitution, calculate the result to see if both sides of the equation remain equal.

Through this substitution, we confirm whether our chosen value solves the equation, or if we need another value instead.

Verification of Solutions

Verification of solutions is the process used to confirm whether a substituted value indeed solves the equation. After substitution and calculation, we need to ensure both sides of the equation match.

• Substitute \(a = 5\) into the equation: \(45 \div 5 = 9\).
• Perform the division to see if the result matches the other side of the equation. In this case, it does: \(9 = 9\).
• If both sides are equal, the solution is verified, confirming \(a = 5\) is correct.

Verification is a critical last step, ensuring no mistakes were made during calculations. It reinforces confidence that the answer is indeed correct.