Problem 69 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ m=\frac{y-b}{x} \quad \text { for } y $$

Step-by-Step Solution

Answer

The solution is \( y = mx + b \).

Step 1: Understand the Problem Structure

We are given the equation \( m = \frac{y-b}{x} \) and need to solve it for the variable \( y \). The equation is in the form of a linear expression divided by \( x \). Our goal is to isolate \( y \) on one side of the equation.

Step 2: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by \( x \). This results in: \( m \cdot x = y - b \).

Step 3: Isolate the Variable \( y \)

Now, to solve for \( y \), add \( b \) to both sides of the equation to move \( b \) to the right side: \( mx + b = y \). This simplifies directly to the expression for \( y \).

Step 4: Verify the Solution

The original expression was \( m = \frac{y-b}{x} \). Substitute \( y = mx + b \) back into this expression to verify: \( m = \frac{(mx+b) - b}{x} = \frac{mx}{x} = m \), which confirms our solution is correct.

Key Concepts

Isolating Variables

Solving equations typically involves a key step called isolating variables, which means getting the variable you're solving for alone on one side of the equation. This allows you to find its value in relation to the other terms. For example, in the exercise with the equation \( m = \frac{y-b}{x} \), isolating the variable \( y \) involves rearranging the equation so that \( y \) is by itself on one side.

• Start by identifying the variable you need to solve for, here it is \( y \).
• Eliminate any fractions or coefficients attached to the variable by performing inverse operations.
• Keep the equations balanced by performing the same operation on both sides.

Once \( y \) has been isolated, you've found an expression that shows exactly what \( y \) is equal to, in terms of the other known values. This process is essential in solving equations across all branches of mathematics.

Linear Expressions

A linear expression is an algebraic expression that represents a straight line graphically. It typically takes the form \( ax + b \), where \( a \) and \( b \) are constants. In our equation, by isolating \( y \), we ended up with the linear expression \( y = mx + b \).

• Linear expressions include variables raised only to the first power.
• They often feature coefficients and constants involved in addition or subtraction, but not multiplication of variables.
• Graphically, these expressions form straight lines.

Understanding linear expressions helps in predicting and simplifying how variables interact. It's foundational in algebra and provides a clear depiction of relationships between variables when graphed.

Algebraic Manipulation

Algebraic manipulation refers to the use of mathematical operations to transform and simplify equations. This is crucial when solving equations like \( m = \frac{y-b}{x} \) for a specific variable.

• First, address any fractions by multiplication to clear denominators.
• Next, perform addition or subtraction to move terms around and away from the target variable.
• Apply any necessary inverse operations such as division or multiplication to further isolate variables.

These steps are not just mechanical procedures; they require understanding which transformations will move you closer to isolating your desired variable. In our specific example, multiplying both sides by \( x \) and then adding \( b \) are algebraic manipulations that facilitated solving for \( y \).