Problem 29 - Algebra A Combined Function

Question

Solve each equation. See Examples 6 and \(7 .\) $$ 4(3 x+2)=12 x+8 $$

Step-by-Step Solution

Answer

The equation has an infinite number of solutions; it's true for all \(x\).

Step 1: Distribute the 4 on the Left Side

Start by distributing the 4 across the terms inside the parentheses on the left side of the equation: \[ 4(3x + 2) = 4 \times 3x + 4 \times 2 \]This simplifies to:\[ 12x + 8 \]

Step 2: Set the Equation

Now that you've distributed the 4, write down the equation with the simplified form on the left side:\[ 12x + 8 = 12x + 8 \]

Step 3: Eliminate Terms to Check for Solutions

Subtract \(12x\) from both sides of the equation:\[ 12x + 8 - 12x = 12x + 8 - 12x \]This simplifies to:\[ 8 = 8 \]

Step 4: Interpret the Result

The resulting equation \(8 = 8\) is always true, indicating that the original equation is an identity. This means the equation is true for all values of \(x\).

Key Concepts

Distribution in Algebra

Distribution is a fundamental concept in algebra that helps simplify expressions and solve equations. It involves spreading a multiplier over terms inside parentheses.
This means you multiply each term inside the parentheses by the term outside.
For example, in the equation \[ 4(3x + 2) \], we distribute the 4 to both terms inside:

• First, multiply 4 by the first term, 3x, to get \( 4 \times 3x = 12x \).
• Next, multiply 4 by the second term, 2, to get \( 4 \times 2 = 8 \).

Thus, \[ 4(3x + 2) \] simplifies to \[ 12x + 8 \].
This simplification makes the equation easier to handle, allowing us to see relationships between different terms. By distributing the multiplication, we cleared the parentheses and can work directly with linear terms.

Solving Identities

An identity in algebra is an equation that holds true for all possible values of the variable involved.
This occurs when simplifying or solving the equation results in a statement that is always true, like \[ 8 = 8 \].
To recognize an identity, compare equations on both sides. If simplifying results in an equivalence like \[ 0 = 0 \], \[ x = x \], or any true numerical equality, then the original equation is an identity.
In our example, the equation \[ 12x + 8 = 12x + 8 \] simplifies to \[ 8 = 8 \] after subtracting \[ 12x \] from both sides.
This is a hallmark of an identity, signifying that no matter the value chosen for \( x \), it will always satisfy the equation. Solving identities can save time in problem-solving as it tells us all solutions apply rather than solving for a specific value.

Equivalent Expressions

In algebra, equivalent expressions refer to different expressions that essentially represent the same value or quantity.
When different expressions simplify to give the same result, they are equivalent.
For the equation \[ 4(3x + 2) = 12x + 8 \], after distribution, both sides of the equation simplify to the same form, \[ 12x + 8 \].
This demonstrates equivalency since the two sides express the exact same relationship.

• To verify equivalency, simplify both expressions separately and compare the results.
• If both expressions yield identical simplified forms, then they are termed equivalent.

Understanding equivalent expressions helps in recognizing and proving the equality of two seemingly different expressions, guiding toward correct solutions and deeper understanding of algebraic properties.