Problem 30 - 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. $$ 5+x^{2}=17 ; x=3 $$

Step-by-Step Solution

Answer

No, x=3 is not a solution to the equation \(5+x^{2}=17\).

Step 1: Substitute the given value

Start by substituting the given value of x (which is 3) into the equation to check if it is a solution. This gives \(5+3^{2}\).

Step 2: Simplify the equation

Simplify the equation by solving \(3^{2}\), which gives 9. This simplifies the equation to \(5+9\).

Step 3: Final simplification & Comparison

Further simplification results in 14. We now compare this result (14) with the value on the other side of the equation (17). If both are equal, then the value of x is a solution to the equation. In this case, however, 14 is not equal to 17, and so the given value of x is not a solution to the equation.

Key Concepts

Substitute Variable in Equation

Understanding the process of substituting a value for a variable in an equation is a fundamental skill in algebra. This technique involves taking a given number, called the substitute, and replacing the variable with it. Let's break down how this applies in our exercise.

When you are presented with an equation like \(5 + x^2 = 17\), and a variable value such as \(x = 3\), the first step is to insert the value of 3 wherever you see the variable \(x\). This transforms the equation into a calculable expression, \(5 + 3^2\), where the exponent here means you will multiply the number 3 by itself. Substitution is like using a placeholder; wherever the variable is, the substitute value will take its place and allow for further calculation. This is essential to solve equations and verify the validity of solutions.

Simplify Algebraic Expressions

The next step in solving an equation after substitution is to simplify the algebraic expression. This means performing any arithmetic operations included in the expression. In our example, after substitution, we obtained \(5 + 3^2\). Simplification is essentially 'cleaning up' the expression to make it as straightforward as possible.

To simplify \(5 + 3^2\), you follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Here, you'd first calculate the exponent, which is \(3^2 = 9\), and then proceed with the addition: \(5 + 9 = 14\). Simplification helps to reduce the expression to its most basic form, making it easier to analyze and solve.

Verify Equation Solutions

Verification is the final and crucial step in checking solutions of equations. It determines whether the substituted value is the correct solution to the given equation. To verify the solution, in our case, you compare the simplified result of the left-hand side of the equation with the right-hand side.

After simplifying \(5 + 3^2\), we got 14. We compare this to the other side of the equation, which is 17. Since 14 does not equal 17, the equation is not balanced, and therefore, \(x = 3\) is not a solution to the equation \(5 + x^2 = 17\). Verification is like the judge at the end of a trial; it confirms whether the proposed solution truly 'fits' the equation, thus ensuring accuracy and understanding of the algebraic process.