Problem 15 - Intermediate Algebra

Question

Use a check to determine whether 5 is a solution of each equation. See Example 1. $$ 3 x+2=17 $$

Step-by-Step Solution

Answer

5 is a solution to the equation because substituting it makes both sides equal.

Step 1: Substitute the Value

Begin by substituting the value 5 into the equation in place of \( x \). This will give you the expression: \( 3(5) + 2 \).

Step 2: Simplify the Equation

Calculate the result of the expression \( 3(5) \). This is \( 15 \), so the equation now becomes: \( 15 + 2 \).

Step 3: Add the Numbers

Add the numbers in the simplified expression: \( 15 + 2 = 17 \).

Step 4: Compare with Original Equation

Check if this result matches the right side of the original equation which is 17. Since it does, this means the value 5 satisfies the equation.

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in algebra to solve equations. It involves replacing variables with given numbers to simplify the equation and verify solutions instantly. This method works by making the equation easier to handle.
To use substitution, you take the given value and "substitute" it into the equation in place of the variable. For instance, in the equation \( 3x + 2 = 17 \), substituting \( x = 5 \) transforms it into a simple arithmetic problem: \( 3(5) + 2 \).
This step makes complex equations more approachable, as it turns them into numerical calculations that are straightforward to solve.

• Identify the variable to substitute.
• Replace every instance of the variable with the value given.
• Carry out any calculations that result from this breakdown.

By mastering substitution, you'll strengthen your skills in handling more complex algebraic expressions.

Solving Equations

Solving algebraic equations involves finding the value of variables that make the equation true. When solving equations, the goal is to isolate the variable on one side of the equation. This can be done through various techniques, among which substitution is a simple yet powerful tool.
In the example we are analyzing, solving \( 3x + 2 = 17 \) requires performing arithmetic operations once substitution has been carried out. These operations are:

• Multiply to evaluate the term with the variable involvement: \( 3(5) \).
• Add the constant term: \( 3(5) + 2 \).
• Assess if both sides of the equation match.

By consistently applying these steps, you can solve equations of varying complexity. Understanding the logical flow will make algebra less daunting and more intuitive.

Solution Validation

Solution validation is an essential step in ensuring the accuracy of algebra solutions. It involves checking that the solution adequately satisfies the original equation.
In our task, once we substitute \( x = 5 \) back into the equation and simplify, we get \( 15 + 2 = 17 \). To validate, we compare this result to the original right-hand side of the equation, which is 17.
If both sides are equal, then the solution is confirmed as correct. Otherwise, reevaluating each step may be necessary.
Important points to consider when validating solutions include:

• Ensure arithmetic operations were correctly performed.
• Double-check substitution accuracy.
• Verify that the simplified equation reflects the original equation.

By incorporating solution validation into your math practice, you safeguard against errors and improve your problem-solving reliability.