Problem 35 - Algebra A Combined Function

Question

Solve each equation. See Examples 6 and \(7 .\) $$ -2(6 x-5)+4=-12 x+14 $$

Step-by-Step Solution

Answer

The equation has infinite solutions, as it is an identity.

Step 1: Distribute the -2

Distribute the -2 to both terms inside the parentheses. The equation becomes:\[ -2 \times 6x + (-2) \times (-5) + 4 = -12x + 14 \]Which simplifies to:\[ -12x + 10 + 4 = -12x + 14 \].

Step 2: Simplify the Left Side

Combine like terms on the left side. The equation becomes:\[ -12x + 14 = -12x + 14 \].

Step 3: Observe the Equality

Notice that both sides are identical after simplification:\[ -12x + 14 = -12x + 14 \]This means the equation is true for all values of \(x\).

Step 4: Conclusion: Determine the Solution

Since the equation holds true for all values of \(x\), this implies there are infinite solutions. In mathematics, this scenario indicates that the equation is an identity.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental algebraic principle that allows you to multiply a single term across terms inside parentheses. This property helps to simplify complex expressions. In this exercise, the distributive property is applied to the expression

• \(-2(6x - 5)\)

which requires distributing

• \(-2\) across
• \(6x\)
• \(-5\)

The expression becomes: \(-2 \times 6x + (-2) \times (-5)\). Notice how we carefully distribute the negative sign along with the number. This is crucial to correctly solving equations and simplifying expressions.

Exploring Infinite Solutions

When both sides of an equation simplify to the same expression, it indicates that the equation has infinite solutions. In our example, after simplifying both sides, we get:

• \(-12x + 14 = -12x + 14\).

This tells us that no matter what value we substitute for \(x\), the equality holds true. Equations like this suggest that the solution set includes all possible real numbers. Therefore, the equation has infinite solutions. This concept is vital when determining the nature of solutions in linear equations.

The Art of Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. It involves merging terms with the same variables to reduce the equation to its simplest form. In our equation, the left side had:

• \(-12x + 10 + 4\)

Here, we combine

• \(10 + 4\)

which simplifies to

• \(14\).

This simplification is important because it helps to clearly see the structure of the equation, revealing that both sides are identical.

Linear Equations as Mathematical Identities

When both sides of an equation are identical post-simplification, it forms what we call a mathematical identity. In this problem, the equation becomes

• \(-12x + 14 = -12x + 14\).

Such identities show us that the equation is true for any value of \(x\). This differs from solving typical equations where only specific values satisfy the equation. Identifying mathematical identities is critical as it informs us about the nature of equations and their solutions. They are especially useful in understanding when an equation does not merely have a single solution, but infinite solutions.