Problem 69 - Algebra A Combined Function

Question

Solve each linear or quadratic equation \(3 x+5=7\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{2}{3}\).

Step 1: Identify the Equation Type

The equation given is a linear equation because it is in the form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

Step 2: Isolate the Variable Term

To solve for \(x\), we first need to isolate the term containing \(x\). Subtract 5 from both sides of the equation to move the constant term to the right side:\[3x + 5 - 5 = 7 - 5\]This simplifies to:\[3x = 2\]

Step 3: Solve for the Variable

Now, divide both sides of the equation by 3 to solve for \(x\):\[x = \frac{2}{3}\].

Key Concepts

Solving Equations

Understanding how to solve equations is fundamental in algebra. An equation is a statement that asserts the equality of two expressions. In mathematics, equations serve as a tool to represent real-world situations using numbers and variables. The goal is to determine the value of the variable that makes the equation true.
When you solve an equation, you often perform operations to isolate the variable on one side of the equation. As you manipulate the equation, it’s vital to perform the same operations on both sides to maintain balance. This approach applies to both linear and quadratic equations.

• Linear equations typically have one variable, with terms not squared or raised to any power greater than one.
• Quadratic equations involve variables that are squared, making them more complex.

Understanding the type of equation you're working with is the first step to determining the appropriate method to solve it.

Variable Isolation

Variable isolation is a crucial step in solving equations. The ultimate goal is to have the variable, often represented by letters like \(x\) or \(y\), alone on one side of the equation. This process involves reversing operations around the variable.
In the equation given, \(3x + 5 = 7\), our primary task is to move everything except the variable to the other side of the equation.
This usually involves a series of inverse operations:

• Start by adding or subtracting terms to remove constants from the variable’s side.
• Once the constant is removed, use multiplication or division to eliminate coefficients (the numbers multiplied by the variable).

For instance, subtracting 5 from both sides removes the constant term, and dividing by 3 allows us to isolate \(x\), resulting in \(x = \frac{2}{3}\). Keeping operations balanced on both sides is vital to correct variable isolation.

Equation Types

Equations can be classified into different types, with linear and quadratic being two of the most common categories in basic algebra. Each type has its characteristics and solving methods.

• Linear Equations: These equations are of the first degree, forming a straight line when graphed. They have the format \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. Solving linear equations is generally straightforward; you perform operations to isolate the variable, as seen in the example \(3x + 5 = 7\).
• Quadratic Equations: These involve variables raised to the second power and follow the format \(ax^2 + bx + c = 0\). Quadratic equations may require methods like factoring, completing the square, or using the quadratic formula.

Identifying the type of equation early on guides you in selecting the appropriate solving strategy, ensuring accuracy and efficiency.