Problem 74 - Algebra A Combined Function

Question

Solve each linear or quadratic equation \(2(3 x+1)+15=-7\)

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Expand the Equation

First, distribute the 2 across the parentheses. This means multiplying 2 by each term inside the parentheses. So, calculate \(2 \times (3x + 1)\). This simplifies to \(6x + 2\). The equation now becomes \(6x + 2 + 15 = -7\).

Step 2: Combine Like Terms

Next, combine the constant terms on the left side of the equation. Add 2 and 15 to get \(17\). Now the equation looks like \(6x + 17 = -7\).

Step 3: Isolate the Variable Term

Subtract 17 from both sides to isolate the term with the variable on one side. This gives \(6x = -7 - 17\). Simplifying the right side, you have \(6x = -24\).

Step 4: Solve for the Variable

Finally, divide both sides of the equation by 6 to solve for \(x\). So, \(x = \frac{-24}{6}\). Simplifying the fraction, you find \(x = -4\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that is used to simplify expressions and equations. When you encounter an expression like \( a(b + c) \), the distributive property allows you to "distribute" the \( a \) across the terms inside the parentheses. This means you apply \( a \) to both \( b \) and \( c \) individually.
For example, in the equation \( 2(3x + 1) + 15 = -7 \), we distribute 2 to both \( 3x \) and 1:

• Multiply 2 by \( 3x \) to get \( 6x \).
• Multiply 2 by 1 to get 2.

The equation then becomes \( 6x + 2 + 15 = -7 \).
By using the distributive property, you open up the expression and make it easier to work with the individual terms, preparing the equation for further simplification.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. It involves adding or subtracting terms that have the same variable raised to the same power. This simplification allows you to work efficiently with equations.
In our example equation, \( 6x + 2 + 15 = -7 \), you look for terms that can be combined:

• The constants 2 and 15 are like terms, as they both do not have a variable component.

Add these constants to simplify the equation. So, 2 plus 15 equals 17, transforming the equation to \( 6x + 17 = -7 \).
Combining like terms helps condense equations, making them neater and often easier to solve by reducing the clutter on one side of the equation.

Isolating the Variable

Isolating the variable is a crucial step in solving linear equations. The goal is to get the variable alone on one side of the equation, which then allows you to determine its value.
With the equation now \( 6x + 17 = -7 \), we need to isolate \( x \). This is done by removing the constant term on the same side as the variable. Subtract 17 from both sides:

• Left side: \( 6x + 17 - 17 \) simplifies to \( 6x \).
• Right side: \( -7 - 17 \) simplifies to \( -24 \).

The equation becomes \( 6x = -24 \).
Next, solve for \( x \) by dividing both sides by 6 to get \( x = \frac{-24}{6} \). Simplifying this gives \( x = -4 \).
Isolating the variable is the step that ultimately allows you to find what value the variable represents in the equation, thus solving it.