Problem 20 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ -5 a=-105 $$

Step-by-Step Solution

Answer

The solution is \(a = 21\).

Step 1: Understand the Equation

The given equation is \[-5a = -105\]. This is a simple linear equation in one variable. Our goal is to solve for \(a\).

Step 2: Isolate the Variable

To find \(a\), we need to isolate it on one side of the equation. This can be done by dividing both sides of the equation by \(-5\). Therefore, \[ a = \frac{-105}{-5} \]

Step 3: Simplify the Division

Simplify the division \(a = \frac{-105}{-5}\) by carrying out the division operation: \[ a = 21\]

Step 4: Verify the Solution

Substitute \(a = 21\) back into the original equation to ensure it satisfies the equation: \[-5(21) = -105\]. Calculate the left side: \[-5 \times 21 = -105\]Both sides are equal, so \(a = 21\) is the correct solution.

Key Concepts

Understanding Linear Equations

A linear equation is a statement of equality involving a linear expression. Linear expressions can include variables, numbers, and addition or subtraction operations. In simple terms, a linear equation is an equation that represents a line when graphed on a coordinate plane.
In the given equation \(-5a = -105\), it is linear because it only involves the variable \(a\), a coefficient \(-5\), and no exponents or powers of variables.

• This type of equation will typically have only one variable, as is the case here.
• The highest power of the variable is 1, making it linear.
• Simplifying such equations often involves basic arithmetic, like addition, subtraction, multiplication, or division.

Linear equations form the basis for many algebraic applications and understanding them is fundamental to more advanced math.

Process of Isolating the Variable

Isolating the variable is crucial for solving any equation, especially linear equations. The main goal here is to have the variable, in this case, \(a\), by itself on one side of the equation.
This process often involves reversing any operations that are currently being applied to the variable. In our exercise, the operation is multiplication by \(-5\). Therefore, the opposite operation, division by \(-5\), will help in isolating \(a\).
Here is how it’s done:

• Start with the equation: \(-5a = -105\).
• To isolate \(a\), divide both sides by \(-5\).
• So, \(a = \frac{-105}{-5}\).

By dividing, you 'cancel out' the \(-5\) on the left side, leaving \(a\) alone. This helps you find the solution in a clear and straightforward way. Consistently practicing this technique builds a strong foundation for solving complex math problems.

Verifying Solutions

Verifying solutions is an essential step to ensure the solution is correct. Once you have derived a possible answer for the variable, in this case, \(a = 21\), it is important to check that this answer satisfies the original equation.
This is done by substituting the value back into the original equation and confirming that both sides are equal. Let’s perform verification for \(a = 21\):

• Substitute \(a = 21\) into the equation: \(-5 \times 21\).
• Calculate the result, which results in \(-105\).
• Since \(-105\) equals the right-hand side of the original equation, \(a = 21\) is indeed the correct solution.

Verification is an important step because it confirms the accuracy of your solution. Additionally, it gives confidence that the procedure used to solve the equation was correct, allowing you to apply the same methods to future problems.