Problem 40 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution is x = 4, and the equation is neither an identity nor a contradiction.

Step 1: Distribute and Simplify

Distribute -3 into the parenthesis: -2x - 3(4 - 2x) = -2x - 12 + 6x On the right-hand side: 2(x-3) + 2 = 2x - 6 + 2

Step 2: Combine like terms

Combine like terms on both sides of the equation: 4x - 12 = 2x - 4

Step 3: Move variable terms to one side

Subtract 2x from both sides to isolate the x-term: 4x - 2x - 12 = 2x - 2x - 4 2x - 12 = -4

Step 4: Isolate the variable

Add 12 to both sides to isolate x: 2x - 12 + 12 = -4 + 12 2x = 8

Step 5: Solve for x

Divide both sides by 2 to solve for x: x = 4

Step 6: Check the solution

Substitute x = 4 back into the original equation to verify the solution: -2(4) - 3(4-2(4)) = 2(4-3) + 2 -8 - 3(4 - 8) = 2 + 2 -8 - 3(-4) = 4 -8 + 12 = 4 4 = 4. The original equation holds true, so x = 4 is a solution.

Step 7: Identify the type of equation

Since there is one solution, the equation is neither an identity nor a contradiction.

Key Concepts

The Distributive Property in Algebra

One foundational concept in algebra is the distributive property, which helps you simplify expressions and equations. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
In mathematical terms, the distributive property is: \[ a(b + c) = ab + ac \].
For example, in the exercise, we are dealing with distribution on the left-hand side:
-\(2 x-3(4-2 x)\).
To apply the distributive property here, we multiply -3 by both 4 and -2x:
\[ -3(4) + (-3)(-2x) = -12 + 6x \].
This helps to open up the parentheses and simplify the equation, making it easier to combine like terms, which is our next step.
Remember:
- Use the distributive property to eliminate parentheses in your equations.
- Carefully multiply each term inside the parentheses by the factor outside.

Combining Like Terms

Once you've applied the distributive property, your next step usually involves combining like terms. In algebra, 'like terms' are terms that have the same variable raised to the same power. For example, 3x and 5x are like terms because both terms contain the variable x raised to the first power.
In our exercise: \[-2x - 12 + 6x = 2x - 6 + 2\],
we can combine -2x with 6x and -12 with -6 and 2:
\[-2x + 6x - 12 = 4x - 12\]
and,
\[2x - 6 + 2 = 2x - 4\].
So after combining like terms on both the left-hand side and the right-hand side, our expression looks simpler:
\[4x - 12 = 2x - 4\].
Combining like terms streamlines our equations and makes it easier to isolate the variable.

Isolation of Variables

The final core concept in solving linear equations is the isolation of variables. This means getting the variable you are solving for all by itself on one side of the equation. This typically involves a few steps:
1. Move variable terms to one side of the equation.
2. Move constant terms to the other side.
3. Simplify the equation to solve for the variable.
Using our example, after combining like terms, we have:
\[4x - 12 = 2x - 4\].
Let's isolate x by moving all x terms to one side and constants to the other:
Subtract 2x from both sides:
\[4x - 2x - 12 = 2x - 2x - 4\],
which simplifies to:
\[2x - 12 = -4\].
Next, add 12 to both sides to isolate 2x:
\[2x - 12 + 12 = -4 + 12\],
which simplifies to:
\[2x = 8\].
Finally, divide both sides by 2:
\[x = 4\].
By isolating the variable, you achieve the simplest form of the problem, making it easy to find the solution.