Problem 73 - 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. \(2 n^{2}+10=14 ; n=1\)

Step-by-Step Solution

Answer

No, \(n=1\) is not a solution to the equation \(2 n^{2}+10=14\).

Step 1: Substitute the given value for the variable

Replace \(n\) in the equation \(2 n^{2}+10=14\) with the given value of 1. So the equation will become \(2(1)^{2}+10\).

Step 2: Simplify the equation

Now, simplify the left-hand side of the equation. \(2(1)^{2}+10 = 2+10 = 12\).

Step 3: Compare to the other side

Next, compare this result to the right-hand side of the equation \(= 14\). In this case, 12 does not equal 14.

Step 4: Make a conclusion

Because 12 does not equal 14, it can be concluded that \(n=1\) is not a solution to the equation \(2 n^{2}+10=14\).

Key Concepts

Substitution Method

The substitution method is commonly used in algebra to determine if a specific value satisfies a given equation. When you substitute, you essentially replace the variable in the equation with the given number.
For instance, in the expression \(2n^2 + 10 = 14\), if you are testing whether \(n=1\) is a solution, you plug \(1\) in wherever you see \(n\). This turns the equation into \(2(1)^2 + 10 = 14\).
By substituting, we change an algebraic expression into a numerical one, making it easier to test if it holds true. Substitution is a direct and efficient way to check potential solutions.

• Identify the variable to substitute.
• Plug in the known value wherever the variable appears in the equation.
• Proceed to simplify to see if the equation balances.

Simplifying Expressions

The process of simplifying an expression in algebra involves reducing it into the most concise form without changing its value. This can involve performing arithmetic operations like addition, subtraction, multiplication, or division.
In our example, when \(n = 1\) is substituted into the equation \(2n^2 + 10\), it results in the numerical expression \(2(1)^2 + 10\).
Simplification follows these steps:

• Calculate the powers or parentheses first: \((1)^2 = 1\).
• Multiply the result by the coefficients: \(2 \times 1 = 2\).
• Add the constants to simplify: \(2 + 10 = 12\).

After simplifying the left side, you can easily see whether it equals the right side of the equation.

Checking Solutions

Once the expression is simplified, the next step is to check if the left side and right side of the equation are equal. This step confirms if the substituted value is a solution to the algebraic equation.
For \(2n^2 + 10 = 14\), after simplifying with \(n=1\), we got \(12\) as the left side, which needs to equal \(14\), the right side.

• Compare both simplified sides of the equation.
• If they are equal, the original substitution was a valid solution.
• If not, as in this case (where \(12 eq 14\)), the value is not a solution.

Checking solutions ensures the reliability of your results and must always be performed to conclude your mathematical findings.