Problem 6 - College Algebra

Question

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

Step-by-Step Solution

Answer

\(x = -\frac{11}{4}\)

Step 1: Identify Like Terms

We have the equation \(7x + 2 = 3x - 9\). We need to move all \(x\) terms to one side and constant terms to the other side.

Step 2: Move x Terms

Subtract \(3x\) from both sides to get the \(x\) terms on one side:\[7x - 3x + 2 = 3x - 3x - 9\] \\[4x + 2 = -9\]

Step 3: Isolate Constant Terms

Subtract 2 from both sides to isolate the \(x\) term:\[4x + 2 - 2 = -9 - 2\] \\[4x = -11\]

Step 4: Solve for x

Divide both sides by 4 to solve for \(x\): \[\frac{4x}{4} = \frac{-11}{4}\] \\[x = -\frac{11}{4}\]

Key Concepts

Solving Equations

Solving linear equations is a foundational skill in algebra that involves finding the value of a variable that makes the equation true. Let's explore the process with our example equation, which is: \[ 7x + 2 = 3x - 9 \]When solving an equation, you always aim to have a setup where the variable, in this case, \( x \), is isolated on one side of the equation. Begin by identifying all the terms with the variable and constant terms. You want to manipulate these terms so that all variables are on one side and constants on the other. Once this setup is achieved, you can perform operations to isolate the variable and solve for its value.In summary, solving an equation might involve several steps of moving terms and performing operations to maintain equality while aiming to isolate the variable of interest.

Algebraic Manipulation

Algebraic manipulation is the heart of solving equations. It's all about using mathematical operations to rearrange terms without changing the equation's equality.How It Works:

• Addition or Subtraction: You can add or subtract the same number (or expression) from both sides of the equation. For example, if you have the equation \(7x + 2 = 3x - 9\), you can subtract \(3x\) from both sides to simplify the expression into \(4x + 2 = -9\).
• Multiplication or Division: Multiply or divide each side of the equation by the same non-zero number. In our example, to solve for \(x\), the equation \(4x = -11\) is simplified by dividing both sides by 4, resulting in \(x = -\frac{11}{4}\).

These operations are key to reshaping the equation into a form where the variables can be isolated and solved. It's important to perform the same operation on both sides to keep the equation balanced.

Isolation of Variables

The ultimate goal when solving equations is the isolation of the variable involved. This means making the variable the subject of the equation. The isolated variable should be on one side of the equation, often on the left, by itself.Step by Step:

• Reorganize Terms: Begin by reorganizing the equation so all terms containing the variable are on one side and constants on the other. In our example \(7x + 2 = 3x - 9\), you move the \(x\)-terms together and the numbers together.
• Perform Operations: Use operations such as addition, subtraction, multiplication, or division to further isolate the variable. In the solution \(4x + 2 = -9\), you subtract 2 from both sides, then divide by 4 to solve for \(x\).

This process requires careful attention to ensure each step follows from the previous one, maintaining balance and eventually leaving you with the solution: in our case, \(x = -\frac{11}{4}\). Successfully isolating the variable confirms you have the correct and final answer.