Problem 1 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using the multiplication property of equality. Be sure to check your proposed solutions. $$\frac{x}{6}=5$$

Step-by-Step Solution

Answer

The solution to the equation \(\frac{x}{6}=5\) is \(x = 30\).

Step 1: Isolate the Variable

The objective is to isolate the variable x on one side of the equation. To accomplish this, multiply both sides of the equation by 6: \(6 * \frac{x}{6}=5 * 6\)

Step 2: Solve for X

Simplify both sides of the equation to find x. On the left-hand side, 6 divided by 6 is 1, so you are left with x. On the right side, 5 times 6 equals 30. So, the result is: \(x = 30\)

Step 3: Check the Solution

Substitute the found value of x back into the original equation to verify if it indeed satisfies the equation. Plugging x = 30 into \(\frac{x}{6}=5\), we get \(\frac{30}{6}=5\), which simplifies to 5 = 5. Since both sides of the equation are equal, the solution is correct.

Key Concepts

Solving Algebraic Equations

Understanding the process of solving algebraic equations is a fundamental skill in mathematics that allows students to find unknown values, generally represented by variables. An equation is a statement that two expressions are equal, and solving the equation means finding the value(s) of the variable(s) that make this statement true.

To solve algebraic equations such as \( \frac{x}{6} = 5 \), one must perform operations that will ultimately isolate the variable, leading to its value being clearly determined. Remember, whatever operation is done on one side of the equation must also be done to the other side to maintain the balance—this is the essence of the multiplication property of equality.

• Identify the form of the equation and the operations involved.
• Perform inverse operations to move closer to isolating the variable.
• Maintain balance by doing the same operation on both sides
• Continue until the variable is isolated and the equation is solved.

This step-by-step method helps students understand the logic behind the operations and reduces the chances of making errors during the process.

Isolating the Variable

Isolating the variable is a crucial step in the solving process, as it involves manipulating the equation to get the variable by itself on one side of the equality. When you encounter an equation like \( \frac{x}{6} = 5 \), your goal is to have 'x' alone on one side.

In the given exercise, the variable 'x' is initially in a fraction. To isolate 'x', you need to eliminate the denominator of '6'. This is where the multiplication property of equality comes into play: multiplying both sides by 6 cancels out the fraction on the left and gives you '6 * 5' on the right.

An important concept to remember is to use 'inverse operations'. In this case, since division by 6 was affecting the variable, we use multiplication by 6, which is the inverse operation, to cancel it out:

• Identify the operation affecting the variable (e.g., division).
• Use the inverse operation to remove it (e.g., multiplication).
• Perform the inverse operation on both sides to maintain equality.

Successfully isolating the variable paves the way to finding the solution for the equation.

Checking Solutions in Algebra

After isolating the variable and arriving at a solution, it's essential to verify its correctness by checking the solution within the original equation. This is an important habit that confirms your result is indeed a solution and helps you to avoid missed errors.

To check the solution, substitute the variable in the original equation with the value you've found. If the equation balances—meaning both sides are equal—then you can be confident in the validity of your solution. For our example, we found that \( x = 30 \). To check it, we substitute '30' for 'x' in the original equation: \( \frac{30}{6} = 5 \). Simplifying the left side, we see that it equals to '5', which is the same as the right side; thus, confirming the solution is correct.

Remember to always:

• Substitute your solution into the original equation.
• Simplify the equation to see if both sides are equal.
• Confirm the authenticity of your solution through this verification step.

This crucial step gives you the assurance that the algebraic manipulations leading to the solution were made correctly.