Problem 23 - Prealgebra and Introductory Algebra

Question

For Exercises 21 to \(32,\) solve for \(y\). $$4 x-y=3$$

Step-by-Step Solution

Answer

\(y = 4x - 3\)

Step 1: Rearrange the equation

Given the equation \(4x - y = 3\), start by moving the term \(y\) to the left-hand side and \(3\) to the right-hand side. This change will look like this: \(4x - 3 = y\).

Step 2: Simplify and Isolate \(y\)

After moving terms around, \(y\) is isolated on one side of the equation, and the equation has been solved for \(y\). Thus, the final solution is \(y = 4x - 3\).

Key Concepts

Linear equations

Linear equations are a type of algebraic equation. They involve variables raised only to the power of one. A linear equation forms a straight line when graphed. The general form is usually written as \(ax + by = c\). Here, \(a\), \(b\), and \(c\) are constants, while \(x\) and \(y\) are variables.
Linear equations have several important characteristics:

• They represent a constant rate of change or a direct relationship between variables.
• Their solutions are generally straightforward and involve basic operations such as addition, subtraction, multiplication, and division.
• The solution represents a line on the Cartesian plane.

Matrices of coefficients and row operations are often used in solving systems of linear equations. In our problem, we dealt with a simple form \(4x - y = 3\), which is a single linear equation in two variables. Understanding the basics of linear equations helps greatly in rearranging and solving these equations.

Rearranging equations

Rearranging equations is a process of changing an equation's form to isolate a specific variable. By doing this, you can simplify one side of the equation and make it easier to work with the variable you are interested in.
To rearrange an equation, you generally need to perform the same operation on both sides to maintain equality.
Here are the common steps for rearranging:

• Identify the term with the variable you want to isolate.
• Move terms involving other variables or constants to the opposite side of the equation by performing inverse operations.
• You may need to combine like terms or simplify expressions in the process.

In the example \(4x - y = 3\), rearranging involves moving the \(y\) term by adding \(y\) to both sides and then subtracting 3 from both sides. This effectively isolates \(y\) on one side, transforming the equation into \(y = 4x - 3\). Rearranging is a foundational skill that's crucial for solving any algebraic equation!

Isolating variables

Isolating variables is an essential step in solving equations. It involves manipulating the equation such that a specific variable stands alone on one side. This allows us to express that variable in terms of other constants and variables present in the equation.
To isolate a variable:

• Perform operations like addition, subtraction, multiplication, or division on both sides.
• Reverse operations surrounding the variable you want to isolate. For example, if a term is multiplied, divide it.
• Simplify any fractions or complex expressions during the process.

The goal is to express the variable as "variable equals expression." In our exercise, this meant moving terms around until \(y\) was by itself, giving the solution \(y = 4x - 3\). Successfully isolating variables is fundamentally important in all levels of mathematics, enhancing your problem-solving skills.