Problem 40 - Algebra A Combined Function

Question

Solve. $$ 9 x+5.5=10 x $$

Step-by-Step Solution

Answer

The solution is \(x = 5.5\).

Step 1: Move Variables to One Side

To begin solving the equation, let's attempt to get all the terms involving \(x\) on one side. Subtract \(9x\) from both sides of the equation:\[9x + 5.5 - 9x = 10x - 9x\]This simplifies to:\[5.5 = x\]

Step 2: Verify the Solution

It's always a good idea to check your solution to ensure it's correct. Substitute \(x = 5.5\) back into the original equation:\[9(5.5) + 5.5 = 10(5.5)\]Calculate both sides:\[49.5 + 5.5 = 55\]\[55 = 55\]Both sides of the equation are equal, confirming that \(x = 5.5\) is the correct solution.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of an unknown variable that makes the equation true. Equations are like balanced scales; whatever you do to one side must be done to the other to maintain equivalence.

To solve an equation, the goal is to isolate the variable, often accomplished by performing operations like addition, subtraction, multiplication, or division on both sides of the equation.

For the equation: \[9x + 5.5 = 10x\] we aim to have all the terms with the variable on one side. This helps to simplify the equation and makes solving it easier.

Variable Isolation

Variable isolation is the process of manipulating an equation to get the variable on one side and everything else on the opposite side.

In our example, \[9x + 5.5 = 10x\], we want to isolate the variable \(x\). This is done by subtracting \(9x\) from both sides: \[9x + 5.5 - 9x = 10x - 9x\]
This simplifies to: \[5.5 = x\] By keeping \(x\) alone, we've effectively solved the equation. Isolating the variable is crucial because it directly leads us to the solution—displaying what \(x\) equals.

Equation Verification

After finding the solution, it's important to verify it to ensure it's correct. This process checks whether the value you found satisfies the original equation.

In our case, substitute \(x = 5.5\) back into the original equation: \[9(5.5) + 5.5 = 10(5.5)\] Calculate the values:

• Left side: \(49.5 + 5.5 = 55\)
• Right side: \(55\)

Both sides of the equation are equal, meaning \(x = 5.5\) is correct. Verification is a critical step in solving equations to confirm the solution and ensure no mistakes were made during calculations.