Problem 70 - Algebra A Combined Function

Question

Solve each linear or quadratic equation \(5 x-1=8\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{9}{5}\).

Step 1: Isolate the variable term

To begin solving the equation \(5x - 1 = 8\), we start by isolating the term containing the variable \(x\). This is done by adding 1 to both sides of the equation to eliminate the constant term from the left side. New equation: \(5x = 8 + 1\)

Step 2: Simplify both sides

Simplify the expression on the right side by performing the addition. This gives us: \(5x = 9\).

Step 3: Solve for the variable

To solve for \(x\), divide both sides of the equation by 5, which is the coefficient of \(x\). This will isolate \(x\) by making its coefficient 1.So, \(x = \frac{9}{5}\).

Step 4: Check the solution

To ensure the solution is correct, substitute \(x = \frac{9}{5}\) back into the original equation \(5x - 1 = 8\) and check if both sides of the equation are equal:Calculate \(5 \times \frac{9}{5} - 1\) = \(9 - 1\)= 8, which is the correct right side of the original equation.

Key Concepts

Isolating the Variable

When solving linear equations like \(5x - 1 = 8\), isolating the variable is a crucial first step. This means you want to have the variable (in this case, \(x\)) by itself on one side of the equation. Usually, this involves simple arithmetic operations to remove constants. In this equation, we start by getting rid of \(-1\) from the left side.

• Add 1 to both sides of the equation: \(5x - 1 + 1 = 8 + 1\)
• The new equation becomes: \(5x = 9\)

The reason we perform the same operation on both sides is to maintain balance. Just like in a balanced scale, whatever you do to one side, you must do to the other. This keeps the equation equality intact.

Simplifying Equations

The next step in solving a linear equation is to simplify it as much as possible. Once we've isolated the variable terms, we work on simplifying the equation. In this particular example, our equation has become \(5x = 9\) after isolating the variable in the previous step.Here, simplifying means checking that each side of the equation is as simplified as possible, which typically involves performing necessary arithmetic operations (like addition, subtraction, multiplication, or division). Since \(5x = 9\) doesn't require further simplification through arithmetic operations, we can move on to solving for \(x\). However, note the simplicity:

• The expression \(5x = 9\) is already simplified, indicating the next step involves solving for \(x\).

Remember, simplified equations are easier to work with and pave the way for finding the solution.

Checking Solutions

Once we find the value for \(x\), checking our solution is essential to confirm its correctness. This is like double-checking your math to make sure you didn't miss anything.In our solved equation, \(x = \frac{9}{5}\). We substitute this back into the original equation \(5x - 1 = 8\):

• Substitute \(x = \frac{9}{5}\): \(5 \times \frac{9}{5} - 1\)
• Calculate: \(9 - 1 = 8\)

Since both sides of the equation are equal after substitution, our solution is verified and correct. Checking your solution helps catch any potential mistakes and ensures your final answer satisfies the original equation.