Problem 61 - Algebra 1

Question

Solve the equation. $$\frac{x}{5}=3$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 15\).

Step 1:Isolation

To solve for \(x\), isolate it on one side of the equation by doing the reverse operation on both sides. In this case, multiply each side by 5. \(\frac{x}{5} \times 5 = 3 \times 5\). This un-does the division by 5 in the original equation.

Step 2:Solving

After multiplying, the equation simplifies to \(x = 15\).

Key Concepts

Isolate Variable

To solve an algebraic equation, such as \( \frac{x}{5} = 3 \), the first step is to isolate the variable. This means we want to get the variable \(x\) by itself on one side of the equals sign. Isolating variables is a fundamental technique in algebra that allows us to find the value of unknown numbers.

In our example, the variable is \(x\), and it is being divided by 5. To isolate \(x\), we need to undo this division by performing the opposite, or inverse, operation on both sides of the equation.

Multiplication as the Inverse of Division

Since division is the operation being applied to \(x\), multiplication is the inverse operation that will cancel out the division. By multiplying each side of the equation by 5, we 'undo' the division and isolate \( x \). You'll see that after this step, the equation will be in the form \(x = \text{some number}\), which is our goal.

Multiplication Operation

The multiplication operation is one of the four basic arithmetic operations and it is used to combine equal groups. When it comes to equations, multiplying both sides of an equation by the same non-zero number keeps the equation balanced and unchanged in value.

For instance, in our example \(\frac{x}{5} = 3\), we perform the multiplication operation by multiplying both sides by 5. By doing this, we ensure the equation remains valid while we simplify it.

How Multiplication Cancels Division

Multiplication by a number is the direct opposite of dividing by that number. This principle allows us to cancel out the division by 5 when we multiply by 5. This results in the simple equation \(x = 3 \times 5\), which can be easily solved. It's critical to understand that the multiplication operation applied in such a manner does not change the solution set of the equation.

Algebraic Equations

Algebraic equations are mathematical statements that express the equivalence of two expressions using a variable. They are fundamental tools used to solve problems and answer questions involving unknown values. The equation \(\frac{x}{5} = 3\) is an example of a simple algebraic equation, where \(x\) represents the unknown value we are trying to find.

Understanding algebraic equations involves recognizing operations and knowing how to manipulate them to obtain a solution. The goal is always to solve for the variable by isolating it.

Solving Algebraic Equations

To solve an equation means finding the value of the variable that makes the equation true. In the given example, after isolating \(x\) and performing multiplication, we determine that \(x = 15\). It is important to check the solution by substituting the value of \(x\) back into the original equation to verify that it satisfies the equation. Successfully solving algebraic equations is a foundational skill in mathematics that has applications across various fields.