Problem 1 - Algebra for College Students

Question

For problems 1-50, solve each equation. \(3 x+4=16\)

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Isolate the Variable Term

Start by moving the constant term to the other side of the equation. Subtract 4 from both sides of the equation: \[3x + 4 - 4 = 16 - 4\]This simplifies to:\[3x = 12\]

Step 2: Solve for the Variable

Now, solve for the variable by dividing each side of the equation by the coefficient of the variable term, which in this case is 3:\[\frac{3x}{3} = \frac{12}{3}\]This simplifies to:\[x = 4\]

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of algebra. In simple terms, solving an equation means finding the value of the variable that makes the equation true.
When you have an equation like \(3x + 4 = 16\), each side of the equation is balanced by the equality sign. So, whatever change you make on one side, you must make the same change on the other side. This keeps the equation valid.

• Identify the variable you need to solve for; in this case, \(x\).
• Move constants (like the number \(4\) in our example) away from the variable term by using inverse operations.
• Continue to isolate the variable until only the variable with its coefficient remains on one side of the equation.

This method ensures that we respect the equality and uncover the solution step by step.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate the variable. In the equation \(3x + 4 = 16\), algebraic manipulation starts by separating the constant from the variable. You do this by performing operations like addition, subtraction, multiplication, or division.
For example:

• Subtract \(4\) from both sides: \(3x + 4 - 4 = 16 - 4\), which simplifies to \(3x = 12\).

This is a crucial skill as it allows you to transform complex equations into simpler ones. By systematically applying operations, you bring the variable closer to being alone, helping you see more clearly what value of the variable solves the equation.

Mathematical Operations

Mathematical operations in algebra are the basic actions we perform on equations to solve them. They include addition, subtraction, multiplication, and division.
These operations help us to both isolate the variable and maintain balance in the equation:

• In step one, subtraction was used to remove \(4\) (a constant) from the variable term \(3x + 4\).
• In step two, division helped us isolate \(x\) by dividing both sides by \(3\) (the coefficient of \(x\)).

Order and consistency in these actions are key because they systematically bring us to the solution. Each operation has an inverse (opposite) which is used strategically to simplify and rearrange terms, ensuring that equations remain properly balanced and solvable.