Problem 61 - Elementary Algebra

Question

Solve. $$ 2 x-(3 x-4)=7-x $$

Step-by-Step Solution

Answer

The equation has no solution.

Step 1: Simplify the Left Side of the Equation

First, let's simplify the left side of the equation by distributing and combining like terms. Start with: \[ 2x - (3x - 4) = 7 - x \] Distribute the negative sign across \[ 3x - 4 \] to get: \[ 2x - 3x + 4 \]. This simplifies to: \[ -x + 4 \].Now the equation is: \[ -x + 4 = 7 - x \].

Step 2: Move Variables to One Side

Next, move all the terms involving \( x \) to one side of the equation.Currently, we have: \[ -x + 4 = 7 - x \].Add \( x \) to both sides to eliminate the \( x \) term on the right:\[ -x + x + 4 = 7 - x + x \].This simplifies to:\[ 4 = 7 \].Notice the \( x \) terms cancel out entirely.

Step 3: Verify the Solution

After simplifying, we end up with \[ 4 = 7 \].This statement is false, indicating that there is no value of \( x \) that will satisfy the original equation. This means the equation has no solution.

Key Concepts

Linear Equations

Linear equations are mathematical expressions involving variables with no exponents greater than one. They often look like this: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. These equations are called "linear" because they form a straight line when graphed on a coordinate plane.

In essence, solving a linear equation means finding the value of the variable that makes the equation true. Here’s a simple way to solve linear equations:

• Isolate the variable on one side of the equation.
• Use arithmetic operations such as addition, subtraction, multiplication, or division.

Linear equations are everywhere in algebra and are foundational to understanding more complex concepts. Remember, always aim to simplify the equation as much as possible to make solving it easier.

Distributive Property

The distributive property is a key algebraic property that helps simplify equations by removing parentheses. It is expressed algebraically as \( a(b + c) = ab + ac \). This principle allows you to distribute multiplication over addition or subtraction.

In our example, we started with \( 2x - (3x - 4) \). Here's how the distributive property works step by step:

• The negative sign in front of the parentheses acts like a \( -1 \) that you distribute to both \( 3x \) and \( -4 \).
• Apply: \( -1 \cdot 3x = -3x \) and \( -1 \cdot -4 = +4 \).

This results in \( 2x - 3x + 4 \), which simplifies further to \( -x + 4 \).
By using the distributive property, you can simplify complex expressions and equations, making it easier to find solutions or, in some cases, determine there isn’t one.

No Solution

When solving equations, sometimes you encounter a situation where no value of the variable will satisfy the equation. This is identified as "no solution."

At the end of our solved equation, we found ourselves with \( 4 = 7 \). This is clearly an incorrect statement. When the variables cancel out and you're left with a false statement like this, it indicates that there is no possible value of \( x \) that will make the equation true.

• A false statement like \( 4 = 7 \) shows inconsistency in the equation.
• No solution occurs often when dealing with equations derived from systems in the real world that have constraints.

Recognizing when an equation has no solution is important. It helps avoid spending unnecessary time seeking an answer where none exists, allowing you to address or re-evaluate the problem.