Problem 49 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(7[2-(3+4 x)]-2 x=-9+2(1-15 x)\)

Step-by-Step Solution

Answer

The equation is an identity, true for all values of \( x \).

Step 1 - Distribute

Distribute the constants inside the brackets on both sides of the equation. For the left-hand side: \[ 7[2-(3+4x)] - 2x = -9 + 2(1-15x) \]Distribute 7 inside the brackets: \[ 7 \times 2 - 7 \times (3 + 4x) - 2x = -9 + 2 - 30x \]Simplify: \[ 14 - 21 - 28x - 2x = -7 - 30x \]

Step 2 - Combine like terms

Combine the like terms on both sides of the equation. \[ 14 - 21 - 28x - 2x = -7 - 30x \] Combine the constants: \[-7 - 30x = -7 - 30x \]Combine the x terms: \[ -30x = -30x \]

Step 3 - Simplify the equation

Notice that the equation \[ -30x = -30x \] is true for all values of \( x \).

Step 4 - Determine the nature of the equation

Since the equation holds true for all values of \( x \), it is an identity.

Step 5 - Check the solution

There is no need for a specific solution check as the equation is an identity.

Key Concepts

Distribution Property

To start solving this equation, we first must use the distribution property. This property allows us to multiply a constant by each term inside the brackets or parentheses.
For instance, in the equation, we distribute 7 across the expression inside the parentheses: \[7[2 - (3+4x)] - 2x = -9 + 2(1 - 15x) \] When you distribute, you get: \[ 7 \times 2 - 7 \times (3 + 4x) - 2x = -9 + 2 - 30x \] This simplifies to:
14 - 21 - 28x - 2x = -7 - 30x.
Always remember to distribute carefully and apply the multiplication to each term inside the parentheses.

Combining Like Terms

After you distribute, the next step is to combine like terms. These are terms that have the same variable raised to the same power or constants (numbers without variables).
From the simplified distribution property example we get:
14 - 21 - 28x - 2x = -7 - 30x.
Combine the constants on the left side:
14 - 21 - 28x - 2x = -7 - 30x
This simplifies to:
-7 - 30x = -7 - 30x.
When combining like terms, always group similar terms and simplify them one step at a time.

Identity Equations

An identity equation is an equation that is true for all values of the variable.
In our example, after combining like terms, we get:
-30x = -30x.
This equation holds true regardless of the value of x.
Thus, it is an identity. Identity equations do not have a specific solution because any value for the variable satisfies the equation.

Contradiction in Equations

Contradiction in equations refers to a situation where no possible value of the variable can satisfy the equation.
An example of a contradiction equation is:
5 = 3.
Clearly, this is impossible and does not hold true under any circumstance.
Unlike identity equations which are always true, contradiction equations are never true.