Problem 28 - Elementary Algebra

Question

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=-\frac{5}{7} \text { and } b=-1 $$

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{5}{7}x - 1 \).

Step 1: Identify Given Values

The problem provides the slope of the line, \( m = -\frac{5}{7} \), and the \( y \)-intercept, \( b = -1 \). These values will be used to write the equation of the line.

Step 2: Understand the Slope-Intercept Form

The slope-intercept form of a line's equation is given by \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the \( y \)-intercept.

Step 3: Substitute the Values into the Slope-Intercept Form

Insert the given values of the slope and \( y \)-intercept into the slope-intercept equation: \( y = -\frac{5}{7}x - 1 \).

Step 4: Simplify if Necessary

Check the equation for any simplification errors. Here, there are none. The equation \( y = -\frac{5}{7}x - 1 \) is already in its simplest slope-intercept form.

Key Concepts

Equation of a Line

When we talk about the equation of a line, we are often referring to mathematical expressions that describe straight lines on a graph. One of the most common forms used to represent this is the "slope-intercept form." This form makes it easier to understand some key characteristics of the line such as its slope and the y-intercept. The general equation for a line in slope-intercept form is written as:\[ y = mx + b \]

• y: Represents the y-coordinate of any point on the line.
• m: Denotes the slope of the line, which is the rate at which y changes for a one-unit change in x.
• x: Represents the x-coordinate of any point on the line.
• b: Is the y-intercept, the point where the line crosses the y-axis.

The elegance of the slope-intercept form is that it instantly provides both the slope and y-intercept values, making graphing and other calculations more straightforward.

Slope

The slope of a line is one of its defining features. It tells you the steepness and direction of the line. In simpler terms, the slope measures how much the line rises or falls as you move from left to right.To calculate the slope, imagine you pick two points on the line:

• The change in y-values between these points.
• Divided by the change in x-values over the same period.
• The result is called "rise over run."

The symbol for slope is often \( m \) and in our example, it is given as \( m = -\frac{5}{7} \). A negative slope, like \(-\frac{5}{7}\), indicates that the line goes downwards from left to right. Hence, if you were to graph this, you'd observe a downward slant.

Y-Intercept

The y-intercept is where the line crosses the y-axis. This point is crucial because it gives a starting value for y when x is zero. In essence, it's telling you what happens when there's no x influence yet; only the y-intercept is visible on the graph.In the slope-intercept form \( y = mx + b \), the y-intercept is represented by the symbol \( b \). In our example, \( b = -1 \), signifying that the line crosses the y-axis at the point (0, -1).Here are things to remember about y-intercepts:

• The y-intercept is always in the form (0, b).
• It is reached when the x-coordinate equals zero.
• On a graph, it's easy to spot as the point where the line touches the y-axis directly.