Problem 124 - College Algebra Essentials

Question

will help you prepare for the material covered in the next section. $$ \text { Solve for } y: 3 x+2 y-4=0 $$

Step-by-Step Solution

Answer

The solution for \(y\) in the given equation is \(y = -\frac{3}{2}x + 2\)

Step 1: Isolate the y-term

The equation given is \(3x + 2y - 4 = 0\). First move all the terms not containing \(y\) to the opposite side of the equation. This will give us \(2y = -3x + 4\)

Step 2: Solve for y

To get \(y\) on its own, divide every term of the equation by 2. This gives us \(y = -\frac{3}{2}x + 2\)

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra, allowing you to find unknown values that make an equation true. In this context, we aim to find the value of a variable, such as \( y \), that satisfies all conditions given in the equation.

When solving equations like \( 3x + 2y - 4 = 0 \), the approach often involves a few simple steps:

• Identify the variable you need to solve for, which is \( y \) in this instance.
• Reorder or manipulate the terms to isolate the variable on one side of the equation.
• Perform operations such as addition, subtraction, multiplication, or division uniformly across all terms to maintain balance.
• Check your solution by substituting back into the original equation to confirm it works.

Remember, the main goal is to rearrange the equation in such a way that the sought variable stands alone.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions or equations to find solutions. This process is essential for solving equations and involves moving terms, combining like terms, and applying arithmetic operations.

In the exercise given, the first step is to move all non-\( y \) terms to the opposite side of the equation, resulting in: \( 2y = -3x + 4 \).

• This typically involves adding or subtracting terms to both sides to move them across the equals sign.
• The next step requires dividing every term by the coefficient of \( y \), which is 2, ensuring \( y \) is isolated: \( y = -\frac{3}{2}x + 2 \).

Mastering algebraic manipulation is crucial as it forms the basis for solving more complex equations and understanding mathematical relationships.

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations. It is written as \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept.

In the solution to \( 3x + 2y - 4 = 0 \), we derived \( y = -\frac{3}{2}x + 2 \), which fits this form:

• The slope \( m \) can help us understand how steep the line is. In this example, \( m = -\frac{3}{2} \) means the line decreases by 3 units for every 2 units it goes horizontally to the right.
• The y-intercept, \( c = 2 \), indicates that the line crosses the y-axis at the point (0, 2).

Understanding slope-intercept form is beneficial for quickly graphing equations and visualizing their behavior. It provides clear insights into both the direction and the point at which the line intersects the y-axis.