Question

Give the equation and graph for a line with y-intercept and slope given in Exercises. $$y \text { -intercept }=-2.5 ; \text { slope }=5$$

Short answer

Answer: The equation of the line is y = 5x - 2.5.

Step-by-step solution

Understand the problem

We need to find the equation of a line given its slope and y-intercept. The slope (\(m\)) is given as \(5\), and the y-intercept (\(b\)) is given as \(-2.5\).

Use the 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. We will plug the given values for slope and y-intercept into this equation.

Substitute the given values

We are substituting the slope, \(m = 5\), and the y-intercept, \(b = -2.5\), into the equation \(y = mx + b\). This gives us the equation: $$y = 5x -2.5$$

Graph the line

To graph the line, we will: 1. Plot the y-intercept, \(-2.5\), on the y-axis. 2. Use the slope to find another point on the line. The slope is the ratio of the change in \(y\) to the change in \(x\) (i.e., rise over run). With a slope of \(5\), this means that for every 1 unit increase in \(x\), \(y\) increases by 5 units. So, if we start at our y-intercept point \((-2.5)\), we can go up 5 units and to the right 1 unit to obtain another point on the line: \((1, 2.5)\). 3. Draw a line through these two points, extending in both directions, to complete the graph. The graph of the line with y-intercept \(-2.5\) and slope \(5\) is a straight line passing through the points \((0, -2.5)\) and \((1, 2.5)\).

Key concepts

Slope-Intercept Form

The slope-intercept form is a quick method to write the equation of a straight line given the slope and y-intercept. The general format is given by the equation
\[ y = mx + b \]
where \( m \) represents the slope and \( b \) stands for the y-intercept. This form is especially useful because it readily provides both the gradient and the starting point of the line on the y-axis.
To illustrate, let's connect it with the problem at hand. Given the slope of \( 5 \) and a y-intercept of \( -2.5 \), one simply substitutes these values into the slope-intercept form to arrive at:
\[ y = 5x - 2.5 \]
This equation is practical because with any value for \( x \), one can instantly solve for \( y \), making it straightforward to graph or to calculate points along the line.

Y-Intercept

Understanding the y-intercept is crucial for graphing and interpreting linear equations. The y-intercept is where the line crosses the y-axis, pinpointing the exact moment when \( x \) equals zero. This feature makes it an invaluable reference point in graphing because you can always begin at a known location on the graph.
With a y-intercept of \( -2.5 \), as in our example, this means that when \( x = 0 \), the value of \( y \) is \( -2.5 \). Therefore, the initial point you would plot on a graph is \( (0, -2.5) \). It signifies the starting height of the line when moving from left to right across the graph. The y-intercept also gives us a glimpse into the function's behavior; in this case, indicating that the line will pass below the origin since the y-intercept is negative.