Problem 40 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 5 a+3=3 $$

Step-by-Step Solution

Answer

The solution is \(a = 0\).

Step 1: Isolate the Variable Term

Subtract 3 from both sides of the equation to isolate the term with the variable on one side. This gives us: \[ 5a + 3 - 3 = 3 - 3 \] Simplifying this, we have:\[ 5a = 0 \]

Step 2: Solve for the Variable

With the equation from the previous step, \(5a = 0\), divide both sides by 5 to solve for \(a\):\[ a = \frac{0}{5} \]This simplifies to:\[ a = 0 \]

Step 3: Check the Solution

Substitute \(a = 0\) back into the original equation to ensure it holds true:Original equation: \[ 5a + 3 = 3 \]Substitute \(0\) for \(a\):\[ 5(0) + 3 = 3 \]This simplifies to:\[ 0 + 3 = 3 \]Which is true: \(3 = 3\). Thus, \(a = 0\) is a correct solution.

Key Concepts

Isolating the Variable

A fundamental step in solving linear equations is to isolate the variable. This means getting the variable of interest, in this case, \(a\), on one side of the equation by itself. This is achieved by performing mathematical operations that eliminate other numbers from that side of the equation.Here's how it's done:

• Start by examining the equation on both sides and identify constants or coefficients that are attached to the variable.
• Use inverse operations to strategically remove these numbers. In our example, we wanted to isolate \(5a\) from the equation \(5a + 3 = 3\). We subtracted 3 from both sides because subtraction is the inverse operation of addition.

After this step, our equation became \(5a = 0\). This transformation brought us closer to having \(a\) by itself, demonstrating how powerful isolating the variable can be in solving equations.

Checking the Solution

Once you find a solution, it's crucial to verify that it's correct by plugging it back into the original equation. This step is called "checking the solution," and it serves as a mathematical double-check.Here's why this is important:

• Verifying ensures that the solution satisfies the original equation, confirming its correctness.
• It helps catch any mistakes made during calculations, offering a chance to correct them.

To check the solution for our problem, substitute \(a = 0\) back into the equation \(5a + 3 = 3\):

• Substitute 0 for \(a\), which gives \(5(0) + 3 = 3\).
• Calculate the expression: \(0 + 3 = 3\), which simplifies beautifully to \(3 = 3\).

Since both sides of the equation are equal, \(a = 0\) is indeed the correct solution.

Simplifying the Equation

Simplifying the equation is an essential step in making the process of solving linear equations more manageable. Simplification involves reducing the equation to its simplest form, often making it easier to see the solution.Here's the approach to simplify equations:

• Perform operations such as addition, subtraction, multiplication, or division to eliminate fractions and combine like terms.
• Always aim to maintain the balance of the equation by performing the same operation on both sides.

In our exercise, we began by subtracting 3 from both sides to simplify the equation:\[ 5a + 3 - 3 = 3 - 3 \]which results in \(5a = 0\). Once simplified, we were able to effortlessly solve for \(a\) by dividing by 5, showing a zero: \(a = 0\). Simplification is a vital strategy in revealing the core solution of an equation with clarity.