Problem 6 - Prealgebra

Question

Check to see if the number to the right of each of the following equations is the solution to the equation. $$5 x-6=9 ; 3$$

Step-by-Step Solution

Answer

Yes, \(x = 3\) is a solution to the equation.

Step 1: Substitute the given value into the equation

We need to check if \(x = 3\) is a solution. Substitute \(x = 3\) into the equation \(5x - 6 = 9\). This gives us \(5(3) - 6 = 9\).

Step 2: Simplify the left side of the equation

Calculate \(5(3)\) which equals \(15\), then subtract \(6\): \(15 - 6 = 9\).

Step 3: Compare both sides of the equation

The result from the left side is \(9\), which matches the right side of the original equation. Hence, both sides are equal.

Step 4: Conclusion

Since both sides of the equation are equal when \(x = 3\), we can conclude that \(x = 3\) is indeed a solution to the equation.

Key Concepts

Substitution Method

The substitution method is a powerful tool for determining solutions to algebraic equations. It involves replacing a variable in an equation with a specific value to check if the equation holds true. This method is particularly useful when you need to verify potential solutions for an equation. Here's how it works:

• Identify the variable you need to substitute. In our case, this is the variable \(x\).
• Take the given value to substitute in place of the variable, such as \(x = 3\) for the equation \(5x - 6 = 9\).
• Replace the variable in the equation with the given number, and simplify the equation to see if both sides are equal.

By substituting \(x = 3\) into the equation, we found that both sides indeed equaled \(9\), confirming our substitution was correct. This method helps verify if a particular number truly solves the equation.

Linear Equations

Linear equations are one of the fundamental structures in algebra. They represent a straight line when graphed on a coordinate plane and are characterized by the expression of a constant and variables raised only to the first power. An example of a linear equation is \(5x - 6 = 9\).Key features of linear equations include:

• Standard form: Typically represented as \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
• Degree: Linear equations are first-degree equations since variables are raised to the power of one.
• Graph: When plotted, a solution of a linear equation lies on a straight line.

For our example, substituting \(x = 3\) resulted in a true statement, illustrating that \(x = 3\) is indeed a solution for the equation \(5x - 6 = 9\). Understanding these properties allows students to recognize when a solution correctly satisfies the equation by making both sides equal upon evaluation.

Solution Verification

Verifying a solution involves confirming that a proposed number satisfies the given equation. This is the final step in proving whether a solution is correct. By taking the original equation, substituting the proposed solution, and simplifying, we can see if the results make logical sense.To perform solution verification effectively:

• Write down the original equation.
• Substitute the given value into the equation, like plugging \(x = 3\) into \(5x - 6 = 9\).
• Simplify the expression to determine if both sides are equal. If they match, the solution is verified.

In our example, by substituting \(x = 3\), both sides of the equation resolved to \(9\), thus verifying that \(x = 3\) is a valid solution. This process ensures the solution's accuracy and transparency.