Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation for \(x\). $$ 7 x+2=3 x-9 $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{11}{4}\).

Step 1: Move Variable Terms to One Side

To solve for \(x\), we first need to group all terms containing \(x\) on one side of the equation. Subtract \(3x\) from both sides: \[ 7x + 2 - 3x = 3x - 9 - 3x \]This simplifies to: \[ 4x + 2 = -9 \]

Step 2: Isolate the Variable Term

Next, we want to isolate the term with \(x\). Subtract 2 from both sides of the equation:\[ 4x + 2 - 2 = -9 - 2 \]This results in:\[ 4x = -11 \]

Step 3: Solve for x

Finally, we solve for \(x\) by dividing both sides by 4:\[ \frac{4x}{4} = \frac{-11}{4} \]Simplifying gives:\[ x = -\frac{11}{4} \]

Key Concepts

Linear Equations

Linear equations are mathematical expressions where each variable is raised to the power of one. They are characterized by expressions like \( ax + b = c \). Here, "\( a \)", "\( b \)", and "\( c \)" represent constants, while "\( x \)" is the variable. These constants are whole numbers or fractions, and their presence means that when plotted, the equation produces a straight line.
Linear equations often model real-world scenarios, such as determining the total cost based on a unit price and quantity. Understanding their foundational structure helps simplify and solve them with efficiency.
Delving into the example equation, \( 7x + 2 = 3x - 9 \), you can observe the linear structure. Knowing this simplifies the process of solving it.

Equation Solving

Equation solving involves finding the value of the variable that makes the equation true. Usually, this requires a series of logical steps to transform an equation into a simpler, more workable form.

• Start by eliminating any constants or similar terms from both sides. This often involves addition or subtraction.
• Next, group terms with the variable on one side for clarity. This step prepares the equation for the final solving stages.

In the original exercise, we initially subtract \(3x\) from both sides. This action helps in grouping the variable terms together. It's a necessary step as it simplifies \(7x + 2 = 3x - 9\) into a more manageable expression.
This sequential approach ensures a logical flow, leading to the ultimate goal of solving for the variable.

Variable Isolation

Variable isolation is the process of simplifying an equation until the variable stands alone on one side. This lets you directly determine its value.
To achieve this, perform operations that remove all coefficients and constants, leaving the variable isolated.

• Use addition or subtraction to eliminate constants from the variable side.
• Then, divide or multiply to get the variable with a coefficient of one.

In the given problem, after simplifying to \(4x + 2 = -9\), we subtracted \(2\) from both sides, reaching \(4x = -11\). The final step is dividing by \(4\), giving us the isolated variable \(x = - \frac{11}{4}\). Each step logically supports the previous one, leading to the desired result.