Problem 42 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(4(2 x+7)=2 x+25+3(2 x+1)\)

Step-by-Step Solution

Answer

The equation is an identity and is true for all values of x.

Step 1: Distribute and simplify

Distribute the constants on both sides of the equation. On the left side: 4(2x+7)=8x+28 On the right side: 3(2x+1)=6x+3 So the equation becomes: 8x+28=2x+25+6x+3 Combine like terms on the right side: 8x+28=8x+28

Step 2: Analyze the equation

Notice that both sides of the equation are identical. 8x + 28 = 8x + 28 Since both sides are the same, the equation is always true, regardless of the value of x.

Step 3: Conclusion

The equation is an identity because it holds true for all values of x.

Key Concepts

Distribution

Distribution is a fundamental step in solving linear equations. It involves multiplying a single term by each term inside parentheses. For example, in the expression \(4(2x + 7)\), distribute the 4 to both \(2x\) and 7:

• 4 \times 2x = 8x
• 4 \times 7 = 28

This yields the expression \(8x + 28\). Similarly, for \(3(2x + 1)\), distribute the 3:

• 3 \times 2x = 6x
• 3 \times 1 = 3

This results in \(6x + 3\). Learning the distribution method is essential because it simplifies complex equations, making them easier to solve.

Simplification

Simplification is the process of making an equation more manageable. After distributing the constants, our equation is:

\[8x + 28 = 2x + 25 + 6x + 3\]

The next step is to combine like terms. Add the x-terms and constants separately:

• Combine the x-terms on the right side: \(2x + 6x = 8x\)
• Combine the constants on the right side: \(25 + 3 = 28\)

The equation now simplifies to:

\[8x + 28 = 8x + 28\]Simplification ensures that the equation is as straightforward as possible, assisting us in identifying the true nature of the equation.

Identity Equations

An identity equation is one that holds true for any value of the variable. In the given problem, after distributing and simplifying, we observe that:

\[8x + 28 = 8x + 28\]

Both sides of the equation are identical, which means no matter what value x takes, the equation will always be true.
Identity equations are important because they confirm that the original equation is valid universally, providing an additional layer of understanding in algebra.

Combining Like Terms

Combining like terms is a crucial step in algebraic simplification. Like terms are terms that contain the same variables raised to the same power. In our equation

\[4(2x + 7) = 2x + 25 + 3(2x + 1)\]

After distributing, we combine the \(2x\) and \(6x\) terms on the right side:

\[2x + 6x = 8x\]

And for the constants:

\[25 + 3 = 28\]

So, we get:

\[8x + 28 = 8x + 28\]

Combining like terms simplifies the equation, making it easier to identify properties like whether it is an identity equation.