Problem 51 - College Algebra with Corequisite Support

Question

Starting with the standard form of an equation \(A x+B y=C\) solve this expression for \(y\) in terms of \(A, B, C\) and \(x\). Then put the expression in slopeintercept form.

Step-by-Step Solution

Answer

The slope-intercept form is \(y = -\frac{A}{B}x + \frac{C}{B}\).

Step 1: Isolate the term with y

Starting with the equation in standard form, \(Ax + By = C\), we first need to isolate the term containing \(y\). To do this, subtract \(Ax\) from both sides of the equation to get \(By = C - Ax\).

Step 2: Solve for y

Next, divide every term in the equation \(By = C - Ax\) by \(B\) in order to solve for \(y\). This gives us \(y = \frac{C}{B} - \frac{A}{B}x\).

Step 3: Put the expression in slope-intercept form

Recall that the slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Rearranging \(y = -\frac{A}{B}x + \frac{C}{B}\), we see that the expression is already in the slope-intercept form with \(m = -\frac{A}{B}\) and \(b = \frac{C}{B}\).

Key Concepts

Standard Form

The standard form of a linear equation is typically represented as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are real numbers, and \( A \) and \( B \) are not both zero. This form is often used because it can easily represent both vertical and horizontal lines. In this setup:

• \( A \) is the coefficient of \( x \).
• \( B \) is the coefficient of \( y \).
• \( C \) is a constant.

This form is beneficial for calculating intercepts and is particularly handy for systems of equations. By keeping equations in standard form, it's simpler to perform addition or subtraction on equations to eliminate variables when solving systems. Here's a basic example: \( 3x + 2y = 6 \), which you can rearrange to find intercepts or slope as needed.

Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is often preferred for graphing because it provides a quick way to determine the starting point and direction of the line:

• \( m \) indicates how steep the line is.
• \( b \) shows where the line crosses the y-axis.

For instance, in the equation \( y = 2x + 3 \), the slope \( m \) is 2, meaning the line rises 2 units for every 1 unit it moves right. The y-intercept is 3, which is where the line crosses the y-axis. This form is derived from the standard form by isolating \( y \), allowing more swift graphing and comprehension of the line's behavior.

Solving for y

To solve for \( y \) means to rearrange the equation in terms of \( y \), essentially making \( y \) the subject of the equation. This process allows us to easily convert to the slope-intercept form:1. Start with the equation \( Ax + By = C \).2. Isolate \( By \) by subtracting \( Ax \) from both sides: \( By = C - Ax \).3. Divide every term by \( B \), giving \( y = \frac{C}{B} - \frac{A}{B}x \).Now \( y \) is expressed entirely in terms of constants \( A \), \( B \), and \( C \), and variable \( x \). This resulting equation, \( y = -\frac{A}{B}x + \frac{C}{B} \), is now in the slope-intercept form, easy to use for plotting graphs and understanding how changes in \( x \) affect \( y \).

Slope

The slope \( m \) of a line signifies its steepness and direction on a graph. In any linear equation in slope-intercept form \( y = mx + b \), \( m \) is the coefficient of \( x \). It tells us how \( y \) changes in response to \( x \):

• If \( m > 0 \), the line ascends as it moves right.
• If \( m < 0 \), the line descends as it moves right.
• If \( m = 0 \), the line is horizontal.

In our example, when converting \( Ax + By = C \) to \( y = -\frac{A}{B}x + \frac{C}{B} \), the slope \( m \) is \(-\frac{A}{B}\). This fraction indicates the vertical change per unit of horizontal change. Understanding slope is crucial in fields like physics and economics, where rate of change is key. It affects everything from speed and distance to trends in data.