Problem 19 - Algebra 1

Question

Rewrite the equation so that \(y\) is a function of \(x .\) $$9-y=1.5 x$$

Step-by-Step Solution

Answer

The rewritten equation is \(y = 9 - 1.5x\).

Step 1: Identify the given equation

The equation given is \(9 - y = 1.5x\). The goal is to transform it into an equation in the form of \(y = f(x)\).

Step 2: Rearrange the equation

To begin isolating \(y\), the equation should be rewritten as \(y = 9 - 1.5x\). Initially, let's subtract \(1.5x\) from both sides of the equation to maintain the equality.

Step 3: Final Equation

After rearranging, \(y = 9 - 1.5x\) is obtained. This is the function of \(x\) in terms of \(y\).

Key Concepts

Equation Rearrangement

Equation rearrangement is a fundamental skill when solving equations or manipulating formulas. It involves changing the structure of an equation to isolate specific variables or express it in a different form. In the exercise given, the goal was to express the equation as a function of \( y \), so \( y \) needed to be isolated on one side of the equation.

To rearrange the given equation \( 9 - y = 1.5x \):

• Identify what needs to be moved around first. The equation start with \( y \) being subtracted from 9, so our aim is to isolate \( y \) on one side.
• Add \( y \) to both sides to get rid of the negative sign and bring \( y \) to the other side.
• Subtract \( 1.5x \) from both sides to shift it to the other side of the equation. This helps in making \( y \) the subject of the equation.

Arranging equations is like solving a puzzle, where each step gently shifts pieces to form the desired picture. Mastering this skill is crucial as it forms the basis for solving more complex problems.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + c \), where \( m \) represents the slope, and \( c \) indicates the y-intercept. This form allows one to see how a change in \( x \) reflects on \( y \).

In our rearranged equation, \( y = 9 - 1.5x \):

• The slope \( m \) is \(-1.5\). This value indicates how steep the line is and in which direction it goes (positive means up, negative means down).
• The y-intercept \( c \) is 9, which shows where the line crosses the y-axis. In practical terms, when \( x = 0 \), \( y \) will be 9.

Understanding the slope-intercept form helps in graphing the equation quickly and appreciating the relationship between \( x \) and \( y \). It's like having a quick overview of the line's behavior without graphing it from scratch.

Isolating Variables

Isolating variables is a key operation in algebra, as it involves rearranging an equation to make a specific variable the subject. This process is immensely helpful for solving equations and understanding relations between variables in mathematical expressions.

To isolate \( y \) from the original equation \( 9 - y = 1.5x \):

• The first step is to remove elements on the same side as \( y \) until \( y \) stands alone. We achieve this by manipulating the equation through addition, subtraction, multiplication, or division.
• In this case, one efficient approach was to add \( y \) to both sides, resulting in \( 9 = 1.5x + y \).
• Subsequently, subtract \( 1.5x \) from both sides, thereby isolating \( y \) as \( y = 9 - 1.5x \).

Although these steps sound straightforward, practicing them often leads to better comprehension and confidence in handling equations. It's a fundamental skill used widely in math and science.