Problem 56 - Algebra 1: Concepts and Skills

Question

Solve \(15 x+6-x=16 x+6-2 x\) $$ \begin{array}{\underline{\phantom{xx}}}{ (A) \text { \(x=0\) }} \quad {(B) \text { \(x=6\) }} \quad{(C) \text { No solution }}\quad {(D) \text { Identity }} \end{array} $$

Step-by-Step Solution

Answer

(D) Identity

Step 1: Simplify both sides of the equation

Combine like terms on both sides of the equation.\nSo, \(15x + 6 - x = 16x + 6 - 2x\) becomes \(14x + 6 = 14x + 6\)

Step 2: Check if the equation is an identity or has no solution

To do this, remove the constants and similar variables on both sides of the equation. If we are left with a false statement (such as 0 = 1), then the equation has no solution. If we are left with a true statement (like 0 = 0), then the equation is an identity.\nSubtract \(14x + 6\) from both sides, we get \(0 = 0\). Thus we end up with a true statement. Hence the given equation is an identity.

Key Concepts

Combining Like Terms

When solving linear equations, the first thing you want to do is simplify each side by combining like terms. Like terms are terms that have the same variable to the same power. In the given exercise, both sides of the equation contain terms involving the variable \(x\).

• The left side is initially \(15x + 6 - x\)
• The right side is \(16x + 6 - 2x\)

To combine like terms, you simply add or subtract the coefficients (the numbers in front of the variables) and keep the variable part the same. So, \(15x - x\) simplifies to \(14x\) on the left side, and \(16x - 2x\) simplifies to \(14x\) on the right side. Both sides also have a constant, \(+6\), which are already simplified.
Hence, when you combine like terms, the equation \(15x + 6 - x = 16x + 6 - 2x\) becomes \(14x + 6 = 14x + 6\). This crucial step simplifies the problem and makes it easier to solve.

Identity Equation

An identity equation is a special case where, after simplifying, both sides of the equation are exactly the same. This means that any value for the variable will satisfy the equation. In the exercise provided, after combining like terms, we get \(14x + 6 = 14x + 6\).
To check if the equation is an identity, we should aim to eliminate all terms involving the variable and constants on both sides. By simplifying further, subtract \(14x + 6\) from both sides, which will give us a statement like \(0 = 0\).

• This results in a trivially true statement, confirming that the original equation is an identity.
• With an identity equation, there is not just one solution—rather, any real number for \(x\) satisfies the equation.

Identifying an identity equation is particularly useful as it helps to recognize when equations have infinite solutions.

Checking for Solutions

Checking for solutions in a linear equation involves verifying if any particular number satisfies the equation.
Once you have simplified the equation by combining like terms, you can test for two primary scenarios:

• If the simplified equation results in a simple true statement like \(0 = 0\) or equivalent, it indicates an identity equation with infinite solutions.
• If, conversely, the simplification leads to a false statement (for instance, \(0 = 5\)), then the equation would have no solutions.

In our example, after simplification, we got \(14x + 6 = 14x + 6\). By further rearranging or subtracting terms from both sides, we ended up with \(0 = 0\). This confirms that our check for solutions test returned a true statement.
Knowing how to check for solutions helps immensely in quickly determining whether you're dealing with an identity equation, a specific solution set, or no solutions at all. It streamlines the process of solving equations and makes sure you understand the nature of the solution you are finding.