Problem 23 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ \frac{x}{4}=7 $$

Step-by-Step Solution

Answer

The solution is \( x = 28 \).

Step 1: Recognize the Equation Type

The equation \( \frac{x}{4} = 7 \) is a simple linear equation where the variable \( x \) is divided by 4.

Step 2: Isolate the Variable

To solve for \( x \), we need to isolate it on one side of the equation. Multiply both sides of the equation by 4 to eliminate the fraction: \[ x = 7 imes 4 \]

Step 3: Perform the Calculation

Carry out the multiplication: \[ x = 28 \] Now, we have found \( x \).

Step 4: Verify the Solution

Substitute \( x = 28 \) back into the original equation to ensure it's correct: \[ \frac{28}{4} = 7 \] Check if the left-hand side equals the right-hand side: 7 equals 7, which confirms our solution is correct.

Key Concepts

Solving Equations

Linear equations are one of the foundations of algebra. They often appear in the form of expressions set equal to each other, such as \( \frac{x}{4} = 7 \). Solving these equations involves finding the value of the unknown variable—in this case, \( x \)—that makes the equation true. To solve a linear equation, it's important to take step-by-step actions that adhere to mathematical operations.

• The first step is to identify the equation type and understand whether it involves addition, subtraction, multiplication, or division.
• In our example, \( x \) is divided by 4, indicating a division operation.

After recognizing the type of linear equation, you aim to eliminate any divisions or multiplications that might be tying the variable down. Solving linear equations is about reversing operations to unveil the unknown.

Isolating Variables

Isolating the variable is a critical phase in solving an equation. This means getting the unknown variable by itself on one side of the equation. For the equation \( \frac{x}{4} = 7 \), the goal is to isolate \( x \).

• The operation applied to \( x \) is division by 4.
• To "undo" this division, multiply both sides of the equation by 4.

By multiplying both sides by the same number, you're balancing the equation, maintaining equality. So, we compute:\[4 \times \frac{x}{4} = 4 \times 7\]This results in eliminating the fraction on the left, simplifying the equation to \( x = 28 \). Always remember that each action taken on one side must be mirrored on the other to keep equations balanced and true.

Checking Solutions

After finding the value of the variable, it's vital to verify that the solution is indeed correct. This involves substitution of the value back into the original equation to check if the equation holds true.

• Substitute \( x = 28 \) back into the original equation: \( \frac{28}{4} = 7 \).
• Simplify the left-hand side: \( 7 = 7 \).

If both sides of the equation are equal, the solution is verified as correct. This step ensures that no mistakes were made during calculations and that the derived answer logically satisfies the equation. Checking solutions not only solidifies your understanding but also builds confidence in solving similar problems in future.