Problem 31 - Algebra 1

Question

Solve the equation. $$8 x=3$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = \frac{3}{8} \)

Step 1: Identify the Equation

The given equation is 8x = 3. x is the variable that needs to be solved for.

Step 2: Isolate the variable x

To isolate x, divide both sides of the equation by 8. This gives x = \( \frac{3}{8} \). This operation is based on the principle that if equals are divided by equals, the quotients are also equal. This principle is essential to keep the equation balanced.

Key Concepts

Variables

In mathematics, a variable is a symbol that represents a number or a value that can change. In the context of linear equations, variables often stand in for unknown numbers that we aim to solve. For example, in the equation \(8x = 3\), the letter \(x\) is the variable. It represents an unknown number that, when solved, makes the equation true.

Variables are essential in algebra as they allow us to express mathematical relationships and solve problems. They act like placeholders, making it easier to apply operations to discover the values they represent. Understanding how to manipulate variables is crucial to solving equations of all types. Just remember that variables simplify expressing complex relationships by allowing manipulation and solution finding.

Equation Balancing

Equation balancing is a vital concept in solving linear equations. It involves making sure that both sides of the equation remain equal as you perform operations to isolate the variable.

Imagine a balance scale where each side must weigh the same. If you add, subtract, multiply, or divide anything to one side, you must do the same to the other side to keep everything balanced. For \(8x = 3\), when we divide both sides by 8, we are actually balancing the equation.

• This operation ensures that we make no change to the overall equality of the equation.
• Keeping an equation balanced allows for the correct solving of the variable, ensuring the integrity of the operation.
• Without balance, the resulting solution would be incorrect and misleading.

Equation balancing emphasizes the equality in an equation and preserves the truth of mathematical relationships.

Division in Equations

Division is a mathematical operation critical to solving equations, especially when isolating variables. In our example \(8x = 3\), we use division to separate the variable \(x\) from its coefficient.

Dividing both sides by the coefficient of the variable \(x\) (which is 8 in this equation) allows us to solve for \(x\). This means we divide 8 by 8 on one side, giving 1, and we divide 3 by 8 on the other side to maintain the balance of the equation:

• The left side: \(8x/8 = x\).
• The right side: \(3/8\).

Thus, \(x = \frac{3}{8}\). By dividing both sides by the same non-zero number, you keep the equation's integrity intact. This simple yet profound technique allows for straightforward solutions to otherwise complicated-looking problems.