Problem 15 - Algebra 1

Question

CHECKING SOLUTIONS OF EQUATIONS Check whether the given number is a solution of the equation. $$5+x^{2}=17 ; 3$$

Step-by-Step Solution

Answer

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

Step 1: Substitute the given value

In the first step, substitute the provided number into the equation in place of \(x\). So, the equation becomes \(5+3^{2}\).

Step 2: Perform the operations

Now calculate \(5 + 3^{2}\). Here, 3^2 is 9. So the equation is now: 5 + 9.

Step 3: Comparison with Right Hand Side

Add 5 and 9 together. We receive \(14\), but on the right hand side of the equation we have \(17\). Since \(14\) is not equal to \(17\), given number 3 is not a solution of this equation.

Key Concepts

Substitution in Equations

Substitution is a key process in algebra used to determine if a number is a solution to an equation. In our example, the equation given is \(5 + x^2 = 17\). The task is to see if 3 satisfies this equation when substituted for \(x\). To start, replace \(x\) with 3 in the equation. This results in the mathematical expression \(5 + 3^2\). Substitution replaces variables with specific numbers so that we can evaluate the equation in concrete terms. After substituting, you'll perform the necessary arithmetic operations to simplify the expression. The substitution process thus helps in checking potential solutions to equations by transforming the abstract into something more tangible.
This is an essential skill needed for solving and understanding algebra problems.

Equations

Equations are mathematical statements that assert the equality of two expressions. In algebra, you typically encounter equations containing variables, which are symbols that represent unknown numbers.Consider the equation given: \(5 + x^2 = 17\). Here, \(x\) is the variable whose value we need to determine. The equation provides information that \(5\) added to \(x^2\) should equal 17. Equations serve as a tool to express relationships and solve for unknown values.

• The left-hand side (LHS) is the expression \(5 + x^2\).
• The right-hand side (RHS) is the value, 17.

The aim is to check if both sides hold true for a particular value of \(x\). If substituting a value for \(x\) makes both sides equal, then it is the solution for the equation.

Algebra 1

Algebra 1 introduces you to the fundamentals of algebraic thinking and operations. It lays the foundation for constructing and solving equations. This involves understanding how to manipulate different algebraic expressions and solve for variables. When checking solutions using substitution, as was done here, Algebra 1 skills allow you to:

• Recognize equations and understand their components (like terms, coefficients, and variables).
• Substitute numbers for variables in equations and simplify them correctly.
• Evaluate expressions by performing order of operations correctly.

Moreover, Algebra 1 teaches you logical thinking and problem-solving strategies, which are crucial in verifying whether potential solutions satisfy the given equations. This problem-solving approach is what makes algebra an essential discipline in mathematics.