Problem 38 - Algebra 2

Question

What are the slope and \(y\) -intercept of the equation \(c x+y=d ?\)

Step-by-Step Solution

Answer

The slope is \(-c\) and the \(y\)-intercept is \(d\).

Step 1: Understand the Standard Form

The given equation is in the form of \( c x + y = d \). This is not the standard slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. The first step is to rearrange the equation to match the slope-intercept form.

Step 2: Rearrange to Solve for y

To convert the equation \( c x + y = d \) into the slope-intercept form, we need to isolate \( y \). Subtract \( cx \) from both sides:\[y = -cx + d\]

Step 3: Identify the Slope

Now that the equation is in the form \( y = -cx + d \), we can identify the slope. The slope is the coefficient of \( x \), which is \(-c\).

Step 4: Identify the y-intercept

Continuing with the equation \( y = -cx + d \), we can identify the \( y \)-intercept. The \( y \)-intercept is the constant term, which is \( d \).

Key Concepts

Understanding the Slope

In the world of linear equations, the slope is a crucial concept as it measures the steepness and direction of a line. It is represented by the letter "m" in the slope-intercept form, which is given by: \( y = mx + b \). The slope tells us how much the value of \(y\) changes for every one unit that \(x\) changes. A positive slope means the line is going upwards as we move to the right, whereas a negative slope indicates the line goes downwards. In our equation, \(y = -cx + d\), the slope is \(-c\). This means that for every increase of 1 in \(x\), \(y\) decreases by \(c\), indicating a downward slant of the line. This understanding is vital for predicting and interpreting the behavior of linear relationships in mathematics and real-world situations.

Defining the y-intercept

The \(y\)-intercept is another fundamental part of linear equations. It refers to the point where the line crosses the \( y \)-axis on a graph. In the slope-intercept form \( y = mx + b \), the \(y\)-intercept is represented by the constant term \(b\). For instance, in the equation \( y = -cx + d \), the \(y\)-intercept is \(d\). This means that when \(x = 0\), \(y\) will be equal to \(d\). It's essentially the starting point of the line when \(x\) is zero and helps in understanding the initial value of \(y\) in the absence of any effect from \(x\). Recognizing the \(y\)-intercept in an equation is useful for graphing lines on the coordinate plane as well as solving real-world problems involving initial conditions or starting values.

Equation Rearrangement Techniques

Rearranging equations is an essential skill in algebra that allows us to manipulate the equation into a more useful form. In order to find the slope and \(y\)-intercept from an equation not initially in slope-intercept form, it is important to know how to rearrange it properly. Consider the equation \(cx + y = d\). Our goal is to transform this into the slope-intercept form, \( y = mx + b \). Here’s how it can be done:

• Isolate \(y\) by subtracting \(cx\) from both sides. \[ y = -cx + d \]
• Now the equation is in the form \( y = mx + b \), where \(-c\) is the slope, and \(d\) is the \(y\)-intercept.

This technique of equation rearrangement is a powerful tool, whether you're adjusting the equation for interpretation, simplification or graphing purposes. Mastering it can greatly enhance your problem-solving skills and mathematical understanding.