Question

In Problems \(35-38\), find the slope and \(y\) -intercept of each line. \(-4 y=5 x-6\)

Short answer

Slope: \(-\frac{5}{4}\), Y-intercept: \(\frac{3}{2}\).

Step-by-step solution

Understand the Linear Equation Form

The standard form of a linear equation is given by \(Ax + By = C\). However, to find the slope and \(y\)-intercept, we need to write the equation in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept.

Rearrange the Equation

Start by rearranging the given equation to solve for \(y\):\[-4y = 5x - 6\]Divide every term by \(-4\) to isolate \(y\):\[y = -\frac{5}{4}x + \frac{3}{2}\]Now the equation is in the form \(y = mx + b\).

Identify the Slope and Y-intercept

In the equation \(y = -\frac{5}{4}x + \frac{3}{2}\), identify the slope \(m\) and the \(y\)-intercept \(b\).The slope \(m\) is \(-\frac{5}{4}\).The \(y\)-intercept \(b\) is \(\frac{3}{2}\).

Key concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is pivotal in understanding the relationship between two variables, typically represented as 'x' and 'y'. This form is noted as \( y = mx + b \). Let's break this down further.

• \( y \) represents the dependent variable, which we solve for.
• \( x \) indicates the independent variable, the known quantity.
• \( m \) is the slope of the line, defining the line's steepness.
• \( b \) stands for the y-intercept, the point where the line crosses the y-axis.

To transition an equation into this form, rearrange it so that \( y \) is isolated. Linear equations are easier to analyze in this form, as you can directly read off both the slope and the y-intercept. This makes it highly efficient for graphing purposes and for easily understanding linear relationships.

Slope of a Line

The slope of a line, often symbolized by \( m \), plays a crucial role in describing the line's direction and steepness. The slope is calculated as the rise over run, which is the change in \( y \) divided by the change in \( x \) between two points on a line.

In simple terms, the slope tells you how much \( y \) changes when \( x \) changes by one unit.

• A positive slope means that the line ascends as it moves to the right.
• A negative slope means that the line descends as it moves to the right.
• A zero slope indicates a horizontal line.
• An undefined slope corresponds to a vertical line.

Understanding the slope is crucial when analyzing trends. For the equation \( y = -\frac{5}{4}x + \frac{3}{2} \), the slope \( m \) is \(-\frac{5}{4}\). This tells us the line decreases by 1.25 units for every unit it moves to the right. The negative sign illustrates that the line is going down as you move from left to right.

Y-Intercept

The y-intercept, denoted by \( b \) in the equation \( y = mx + b \), marks the point where the line intersects the y-axis. Understanding the y-intercept is essential in quickly locating a line on a graph.

It is important because:

• The y-intercept is where the value of \( x \) is zero.
• It provides a starting point for graphing a line on the coordinate plane.
• It is particularly useful for predicting outcomes when \( x \) isn't changing.

In our example equation \( y = -\frac{5}{4}x + \frac{3}{2} \), the y-intercept \( b \) is \( \frac{3}{2} \). This means that when \( x \) is zero, \( y \) equals \( \frac{3}{2} \). Hence, the graph of this line crosses the y-axis at the point \( (0, \frac{3}{2}) \). Recognizing this point allows for a clear visual starting point for sketching the entire line.