Problem 9 - Algebra 1

Question

Solve the equation. $$-3 x=5$$

Step-by-Step Solution

Answer

The solution to the equation \(-3x = 5\) is \(x = - \dfrac{5}{3}\).

Step 1: Interpret the Equation

The equation given is \(-3x = 5\). This is a simple linear equation where we have to solve for \(x\).

Step 2: Isolate the Variable

To isolate \(x\) on one side of the equation, the \(3\) must be moved on the other side. This is done by dividing both sides of the equation by \(-3\). Doing so, the equation becomes \(x = - \dfrac{5}{3}\). This is done because it is known that division is the opposite of multiplication. As \(-3\) was multiplied with \(x\), it needs to be divided from \(5\) to balance the equation.

Step 3: Verify the Solution

To verify if \(x = - \dfrac{5}{3}\) is a solution, substitute \(x\) with \(- \dfrac{5}{3}\) in the original equation. The left-hand side becomes \(-3 * - \dfrac{5}{3} = 5\), which is equal to the right-hand side. Therefore, the solution satisfies the original equation.

Key Concepts

Isolating the Variable

A linear equation seeks to find the value of a variable, usually represented by letters like \(x\). The goal is to isolate this variable. In our equation, \(-3x = 5\), the term \(-3x\) shows \(x\) being multiplied by \(-3\).
To isolate \(x\), we need to "undo" this multiplication by \(-3\). Fortunately, in mathematics, division acts as the opposite operation of multiplication. Therefore, we divide both sides of the equation by \(-3\):

• The left side becomes \(x\), as \(-3x\) divided by \(-3\) leaves \(x\) alone
• The right side becomes \(-\frac{5}{3}\), since \(5\) divided by \(-3\) results in a fraction

This process of isolating the variable helps us see clearly that \(x = -\frac{5}{3}\) is the potential solution.

Verifying Solutions

After finding the potential value of \(x\), it’s crucial to determine if it truly solves the equation. Verification assures us that our solution is correct. In our case, we found \(x = -\frac{5}{3}\).
Substitute \(x\) back into the original equation to check: \(-3\left(-\frac{5}{3}\right) = 5\).

• Here, multiplying \(-3\) by \(-\frac{5}{3}\) should return \(5\).
• The negative signs cancel each other during the multiplication, making the expression positive.
• Since \(\frac{15}{3}\) simplifies cleanly to \(5\), the equation holds true.

This process verified that our solution is correct, showing \(x = -\frac{5}{3}\) satisfies the equation.

Basic Arithmetic Operations

Solving equations requires understanding basic arithmetic operations like addition, subtraction, multiplication, and division. These operations provide the tools to manipulate equations and uncover the variable's value.
In our exercise \(-3x = 5\), the key operation is division.

• Understanding multiplication and division as opposite operations helps to rearrange equations effectively.
• This example used division to isolate \(x\) because it involved multiplying by \(-3\).

Arithmetic operations work together like a team, allowing you to correctly solve equations by balancing both sides of the equation and isolating variables. Mastery of these basics ensures a strong foundation for solving any linear equation.