Problem 22 - Algebra 1

Question

Is the number given a solution of the equation? $$5 x+2=17 ; 3$$

Step-by-Step Solution

Answer

Yes, 3 is a solution to the equation \(5x + 2 = 17\).

Step 1: Identify the equation and given number

The given equation is \(5x + 2 = 17\) and the number to check if it's a solution is 3.

Step 2: Substitute the given number into the equation

Replace \(x\) in the equation with 3: \(5*3 + 2\)

Step 3: Simplify the left-hand side of the equation

Perform the multiplication and addition to simplify: \(5*3 = 15\), and add 2 to get 17.

Step 4: Compare the simplified equation to the original equation

The simplified equation is \(17 = 17\), which is equivalent to the original equation, indicating that \(x = 3\) is indeed a solution.

Key Concepts

Solution Verification

To verify a solution means to check if a certain number satisfies an equation. We will explore how you verify solutions through a systematic approach. In the context of algebraic equations, such as \(5x + 2 = 17\), confirming whether a solution is correct requires substituting the specific value back into the equation. Here's the process:

• First, identify the number you want to verify as a potential solution.
• Substitute this number for the variable within the given equation.
• Simplify the equation to see if both sides are equal.

Once you've gone through these steps, if both sides of the equation are equal, the number is a verified solution. In our example, we found \(x = 3\) to satisfy the equation, confirming that 3 is indeed a solution.

Substitution Method

The substitution method is the process of replacing variables with numbers to check potential solutions in equations. In algebra, this method is pivotal when solving for unknowns. Let's break down how this is done in our case:

• Identify the equation: \(5x + 2 = 17\).
• Define the variable replacement: we replace \(x\) with 3.
• Perform the substitution: alter the equation to reflect the replacement. So it becomes \(5(3) + 2\).

After substitution, solve the altered equation. This results in an expression that can be simplified, helping to verify if the value is a correct solution.

Simplifying Expressions

Simplifying expressions is a fundamental step in solving and verifying algebraic equations. To simplify means to perform operations to condense an expression down to its simplest form. Here's how it applies to our situation:Start by addressing the numerical side of the equation:

• Execute multiplication first: \(5 \times 3 = 15\).
• Next, add the remaining numbers: \(15 + 2 = 17\).

Once the left side of the equation equals the simplified form, compare it to the right side. If both sides are alike, the simplification confirms the substitution outcome. For \(5x + 2 = 17\), simplifying confirms that \(x = 3\) is a valid solution because the equation balances as \(17 = 17\).