Problem 28 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(2(3-2 x)=x-4\)

Step-by-Step Solution

Answer

The solution is \(x = 2\). The equation is neither an identity nor a contradiction.

Step 1 - Distribute the 2

Distribute the 2 on the left side of the equation: \[2(3-2x) = x - 4\]This results in: \[6 - 4x = x - 4\]

Step 2 - Move variable terms to one side

Add 4x to both sides of the equation to move all variable terms to one side: \[6 - 4x + 4x = x - 4 + 4x\]This simplifies to: \[6 = 5x - 4\]

Step 3 - Isolate the variable term

Add 4 to both sides to isolate the term with the variable: \[6 + 4 = 5x - 4 + 4\]Which results in: \[10 = 5x\]

Step 4 - Solve for x

Divide both sides by 5 to solve for \(x\): \[\frac{10}{5} = \frac{5x}{5}\]This simplifies to: \[x = 2\]

Step 5 - Check the solution

Substitute \(x = 2\) back into the original equation to verify: \[2(3 - 2*2) = 2 - 4\]This results in: \[2(3 - 4) = -2\]Which simplifies to: \[2(-1) = -2\]Since both sides are equal, \(x = 2\) is a correct solution.

Step 6 - Determine identity or contradiction

Since the equation has a specific solution \(x = 2\), it is neither an identity nor a contradiction.

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps simplify expressions. It states that: \[ a(b + c) = ab + ac \]
This means you distribute the multiplication over addition or subtraction within the parentheses.
In our example: \[ 2(3 - 2x) = x - 4 \]
We apply the distributive property to get: \[ 6 - 4x = x - 4 \] Notice how the 2 is multiplied with both 3 and -2x inside the parentheses.

Isolating Variable

Isolating the variable is a crucial step to solve an equation. The goal is to get the variable alone on one side of the equation. This makes it easier to see the solution.
In our example: \[ 6 - 4x = x - 4 \]
We want to move all x terms to one side.
First, add 4x to both sides: \[ 6 - 4x + 4x = x - 4 + 4x \]
This simplifies to: \[ 6 = 5x - 4 \]
Now, add 4 to both sides to isolate the term with the variable: \[ 6 + 4 = 5x - 4 + 4 \]
Which results in: \[ 10 = 5x \]

Checking Solutions

After solving an equation, always check your solution. Plug it back into the original equation to verify.
For our solution, x = 2, we substitute it back into the original equation: \[ 2(3 - 2*2) = 2 - 4 \]
This simplifies to: \[ 2(-1) = -2 \]
Both sides are equal, which confirms our solution is correct.
This step ensures that no mistakes were made during the computation.

Identity vs. Contradiction

Understanding the difference between an identity and contradiction is vital.
An identity is an equation that is true for all values of the variable. For example: \[ x + 0 = x \]
A contradiction is an equation that has no solution. For example: \[ x + 1 = x \]
In our example, since we found a specific solution (x = 2), it's neither an identity nor a contradiction.
The equation has a unique solution, making it a standard linear equation.