Problem 67 - Algebra A Combined Function

Question

\(10=2 x-1\)

Step-by-Step Solution

Answer

x = 5.5

Step 1: Isolate the Variable

First, you want to get the variable 'x' by itself. Start by adding 1 to both sides of the equation to eliminate the -1 on the right side:\[10 + 1 = 2x - 1 + 1\]This simplifies to:\[11 = 2x\]

Step 2: Solve for the Variable

Now, you need to solve for 'x' by getting rid of the coefficient of 2 that is in front of 'x'. Do this by dividing both sides of the equation by 2:\[\frac{11}{2} = \frac{2x}{2}\]This gives:\[x = \frac{11}{2}\] or \[x = 5.5\]

Key Concepts

Solving Equations

Solving equations is a foundational skill in algebra, where the goal is to find the value of the unknown variable. In our original exercise, we have an equation:\[10 = 2x - 1\]To solve it, we need to perform a series of logical steps that will reveal what 'x' is.
Solving equations often involves multiple operations like addition, subtraction, multiplication, or division, applied to both sides of the equation. This ensures we maintain the equation's balance. Here,we first eliminated subtraction by adding, and then used division.
These steps help gradually reduce the equation into a simple statement like \(x = \ldots\), revealing the solution.
A systematic approach to solving ensures you don't miss crucial operations, thus making problem-solving more efficient.

Isolation of Variable

Isolation of variable is a critical step in solving equations. It involves rearranging the equation so that the unknown variable stands alone on one side of the equation.In our example \(10 = 2x - 1\), we aimed to have 'x' by itself on one side.
The initial step was to address any numbers or coefficients attached to 'x'. This was done by eliminating the \(-1\) by adding \(1\) to both sides of the equation, resulting in:\[11 = 2x\]This process of isolating ensures that we systematically simplify the equation, making it clearer what remains to be solved.
This step sets the stage for the final calculations needed to uncover \(x's\) true value.

Algebraic Manipulation

Algebraic manipulation refers to the process of using arithmetic operations to transform an equation into a more workable form. This transformation helps us more easily isolate variables or simplify expressions.In our exercise, once we isolated 'x' by moving constant terms around, we performed algebraic manipulation to remove the coefficient in front of it.
The step involved dividing both sides by \(2\), allowing us to simplify the expression \(11 = 2x\) to \(x = \frac{11}{2}\), or \(x = 5.5\).
Manipulating equations not only involves basic operations but also understanding the properties of equality, ensuring any steps taken mirror each other on both sides of the equation.This methodical approach in manipulation is vital towards reaching the correct and simplest form of a solution.