Question

Write the equation of each straight line and make a graph. Slope \(=-1.50 ; y\) intercept \(=3.70\)

Short answer

The equation of the line is \(y = -1.50x + 3.70\). To graph it, start at the point (0, 3.70) on the y-axis and use the slope to find other points on the line.

Step-by-step solution

Understanding the Slope-Intercept Form

The slope-intercept form of the equation of a straight line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Knowing the slope and y-intercept allows us to write the equation directly.

Insert the Given Slope and y-intercept

Insert the given slope \(-1.50\) and y-intercept \(3.70\) into the slope-intercept form equation. The equation becomes \(y = -1.50x + 3.70\).

Graphing the Equation

To graph the equation \(y = -1.50x + 3.70\), start at the y-intercept \(3.70\) on the y-axis. Then, use the slope to determine the next point. Since the slope is \(-1.50\), from the y-intercept, go down 1.50 units vertically for every 1 unit you go horizontally to the right (since the slope is negative). Connect these points to draw the line.

Key concepts

Linear Equations

Linear equations are fundamental in the world of algebra and act as a starting point for understanding more complex mathematical concepts. They have the unique characteristic of forming straight lines when graphed on a coordinate plane. A linear equation in two variables, like x and y, can be expressed in various forms, with the most recognizable being the slope-intercept form: \( y = mx + b \).

In this equation, \( m \) represents the slope of the line, which indicates its steepness and direction. The \( b \) stands for the y-intercept, which is the point where the line crosses the y-axis. The values of \( m \) and \( b \) determine the specific position and angle of the line on the graph. It is crucial to comprehend these components to effectively write and graph linear equations.

An equation like \( y = -1.50x + 3.70 \) can tell us a lot at a glance; the line falls downwards as it moves from left to right since the slope is negative (-1.50), and it crosses the y-axis above the origin, at 3.70. Understanding linear equations in this form gives us the power to quickly picture their graphs in our minds.

Graphing Linear Equations

Graphing is a visual representation that helps us to better understand the behavior of linear equations. The graph of a linear equation results in a straight line, and all it takes to draw this line are two points, which can be located using the slope and the y-intercept from the slope-intercept form of the equation.

Beginning at the y-intercept provides the first point; from here, apply the slope rule \( \text{rise over run} \) to find a second point. If the slope \( m = -1.50 \), we interpret this as a rise (movement in the y-direction) of -1.50 for each unit we run (move in the x-direction). A negative rise means we go down when moving from left to right, establishing the downward angle of our line.

After plotting these points on a coordinate grid, simply draw a line through them, which extends infinitely in both directions. This line is the graphical representation of the equation and includes all the possible solutions (x, y) that satisfy the given linear equation.

Slope and Y-intercept

The slope and y-intercept are crucial components that define the characteristics of a line's graph. The slope, denoted \( m \), is a measure of how steep the line is. A positive slope indicates that the line rises as it moves from left to right, while a negative slope means it falls. The steeper the line, the greater the absolute value of the slope.

Meanwhile, the y-intercept, represented by \( b \), is the point where the line crosses the y-axis. This value tells us where to begin plotting our line on the graph. Together, these two values make it easy to not only graph a line but also to understand the directional flow and starting point of the line.

A quick tip: if you're ever in doubt about the direction of the line, remember that a positive slope slants 'up', and a negative slope slants 'down'. And regardless of the slope, the line will always pass through the y-axis at the point \( (0, b) \). Armed with these insights, the task of plotting any linear equation should be straightforward and approachable.