Problem 33 - Algebra 1: Concepts and Skills

Question

CHECKING SOLUTIONS OF EQUATIONS Check to see if the given value of the variable is or is not a solution of the equation. \(n^{2}-5=20 ; n=5\)

Step-by-Step Solution

Answer

Yes, \(n = 5\) is a solution to the equation \(n^2 - 5 = 20\).

Step 1 Substitute

Plug \( n = 5 \) into the given equation \(n ^2 - 5 = 20\). We have to replace \( n \) with 5 in the equation and see if both sides of the equation are equal.

Step 2 Compute

Calculate the left side of the equation. With \( n = 5 \) the left side of the equation becomes \(5^2 - 5 = 25 - 5 = 20.\) Here, the left and right sides of the equation are equal, hence \( n = 5 \) is indeed a solution of the given equation.

Key Concepts

Substitution Method

The substitution method is an essential tool to help students confirm the solutions to algebraic equations. This method involves replacing the variable in an equation with a given value to see if the equation holds true. Let's relate this to our example involving checking the solution of an equation.

When given the equation \(n^{2}-5=20\) and the value \(n=5\), the substitution method requires you to replace the variable \(n\) with 5 in the equation. If after performing the operations on the left side of the equation, the result is equal to the right side, then \(n=5\) is a correct solution. This method is particularly handy as it serves as a quick check, confirming whether a proposed solution is correct or not. It is a fundamental approach for solving algebraic equations that also builds a strong foundation for more complex problem-solving strategies.

Solving Algebraic Equations

Solving algebraic equations is a process that involves finding the value of the variable that makes the equation true. This skill is integral to mathematics and requires understanding various techniques. One such technique is the substitution method mentioned earlier. However, there are also processes like factoring, using the quadratic formula, graphing, and completing the square.

A good first step is to simplify the equation, if possible, by combining like terms and using inverse operations to isolate the variable. For instance, if an equation includes \(n^{2}\), we'd look to move all other terms to the opposite side of the equation to isolate the squared variable. Doing this ensures that students can apply appropriate methods to find the correct value of the variable that satisfies the equation.

Quadratic Equations

Quadratic equations are a special type of algebraic equation where the highest degree of the variable is two, written generally as \(ax^{2} + bx + c = 0\). Solving quadratic equations is a key topic in algebra that often involves methods like factoring, using the quadratic formula, completing the square, or graphing.

In our textbook example, the equation \(n^{2}-5=20\) can be identified as a quadratic equation because it can be rewritten in the standard quadratic form \(n^{2} - 5 - 20 = 0\). Solving this particular quadratic equation can be straightforward because it is already in a form that makes it easy to apply the square root method after moving all terms to one side and isolating \(n^{2}\). Understanding how to manipulate and solve quadratic equations is crucial because these equations appear frequently in both academic studies and real-life applications.