Problem 31 In questions $25-47$ solve eac... [FREE SOLUTION]

Chapter 7: Problem 31

In questions $25-47$ solve each equation: $$ 2 x-1=x-3 $$

Short Answer

Expert verified

The solution is $ x = -2 $.

Step by step solution

Simplify the Equation

Begin solving by moving all terms involving the variable to one side of the equation. Subtract $ x $ from both sides, which eliminates $ x $ from the right side.\[ 2x - x - 1 = x - x - 3 \]This simplifies to:\[ x - 1 = -3 \]

Isolate the Variable

Next, eliminate the constant term on the side with the variable. Add $ 1 $ to both sides of the equation to isolate $ x $.\[ x - 1 + 1 = -3 + 1 \]This simplifies to:\[ x = -2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable isolation

When solving a linear equation, the goal is to determine the value of the variable that makes the equation true. Variable isolation refers to the process of getting the variable by itself on one side of the equation.In our example equation, after simplifying to get $ x - 1 = -3 $, we focus on isolating $ x $ by undoing operations around it. To do this:

Notice that $ -1 $ is attached to $ x $. To isolate $ x $, we need to eliminate $ -1 $ by doing the opposite mathematical operation which is addition.
Add $ 1 $ to both sides of the equation. This step cancels out the $ -1 $ next to $ x $ on the left-hand side of the equation: $ x - 1 + 1 = -3 + 1 $.

Once the constant is eliminated, $ x $ stands alone, and we have a clearer picture of the variable's value which is $ x = -2 $. Successfully isolating the variable is a critical skill as it sets the foundation for solving the equation.

Equation simplification

Equation simplification involves reducing the equation to its simplest form. This means performing operations that combine like terms and eliminate any redundancy, which makes solving for the variable more straightforward.In the original exercise, the equation starts as $ 2x - 1 = x - 3 $. We simplify this by:

Moving all expressions involving the variable $ x $ to one side. Subtract $ x $ from both sides of the equation: $ 2x - x - 1 = x - x - 3 $.
After subtracting, we combine like terms to simplify the expression: $ x - 1 = -3 $.

Simplification helps in managing the complexity of the equation by reducing it to a form that's easier to handle in the next steps of solving, such as isolation of the variable.

Solving for x

Solving for $ x $ means finding the value of $ x $ that makes the equation true. Once we've simplified the equation and isolated the variable, we directly solve for $ x $.After simplifying and isolating in our example, we ended with the equation $ x = -2 $. This means:

$ x = -2 $ is the solution since substituting it back into the original equation should hold the equation true.
Verify the solution by substitution: Replace $ x $ in the original equation $ 2x - 1 = x - 3 $ with $-2$:

For verification:

Calculate the left side: $ 2(-2) - 1 = -4 - 1 = -5 $.
Calculate the right side: $-2 - 3 = -5$.
Both sides equal $-5$. Thus, $ x = -2 $ is indeed the correct solution.

Solving for $ x $ helps confirm the validity of the operations performed in previous steps and ensures the solution correctly satisfies the original equation.

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91影视

In questions \(25-47\) solve each equation: $$ 2 x-1=x-3 $$

Short Answer

Step by step solution

Simplify the Equation

Isolate the Variable

Key Concepts

Variable isolation

Equation simplification

Solving for x

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