Question

Graph the line that passes through the points. Write its equation in slope- intercept form. (Review \(5.3 \text { for } 5.7)\) $$(-5,-2),(-1,12)$$

Short answer

The line that passes through the points (-5,-2) and (-1,12) has an equation of \(y = 3.5x + 15.5\)

Step-by-step solution

Identify and Plot the Points

First, to get the idea of overall graph, plot the given points on the coordinate plane. Here the points are (-5, -2) and (-1, 12)

Calculate Slope

The formula for slope given two points (x1, y1) and (x2, y2) is \((y2 - y1) / (x2 - x1)\). Here, (x1, y1) is (-5, -2) and (x2, y2) is (-1, 12). Substituting these into the formula, the slope (m) becomes \((12 - (-2))/(-1 - (-5)) = 14/4 = 3.5\)

Write Equation

Now that we have the slope, we can use the slope-intercept form y = mx + b, where m is 3.5 and (x, y) could be either of the given points. Let's use the first point (-5, -2): substituting into the equation gives us \(-2 = 3.5*(-5) + b\). Solving for b we find \(b = 15.5\)

Final Equation and Graphing

Now that we have the y-intercept (b=15.5) and the slope (m=3.5), the final equation is \(y = 3.5x + 15.5\). To graph this, start at the y-intercept (0,15.5) and use the slope (rise over run) to draw the line. It will pass through the two original points (-5, -2) and (-1, 12).

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for graphing linear equations. It is written as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) represents the y-intercept, the point where the line crosses the y-axis. In the given exercise, with the slope (\(m\)) calculated as 3.5 and the y-intercept (\(b\) found to be 15.5, the linear equation becomes \( y = 3.5x + 15.5 \) which is in the slope-intercept form.

This particular form is especially helpful for graphing because it gives a clear starting point (the y-intercept) and a consistent way to move along the line (the slope). For anyone trying to plot a line on a coordinate plane, knowing these two values can simplify the process significantly.

Slope of a Line

The slope of a line measures its steepness, commonly described as 'rise over run'. To calculate it, take any two points on the line; for instance, in our exercise, the points are \( (-5, -2) \) and \( (-1, 12) \). By applying the slope formula \( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \), we can determine the line's slope.

For these points, the slope is \( 3.5 \), indicating for every 3.5 units the line rises vertically, it runs 1 unit horizontally. Slope is vital because it dictates the direction and angle of the line on the graph. A positive slope means the line tilts upwards, while a negative slope means it tilts downwards. Understanding slope helps students conceptualize how different linear equations can look on a graph.

Coordinate Graphing

Coordinate graphing refers to the process of plotting points, lines, and shapes on a two-dimensional plane to represent equations or data visually. Each point is determined by an ordered pair of numbers \( (x, y) \), which correspond to horizontal (x-axis) and vertical (y-axis) locations.

For example, in the exercise given, the points \( (-5, -2) \) and \( (-1, 12) \), represent specific locations on the graph. To graph the line from the equation \( y = 3.5x + 15.5 \), you would start at the y-intercept \( (0, 15.5) \) and use the slope to find another point by 'rising' 3.5 units up for every unit you 'run' to the right. Repeating this process allows you to draw the precise line. Graphing these elements helps students visualize the behavior of linear equations, enhancing their understanding of algebraic concepts.

Utilizing the coordinate plane effectively can turn abstract algebra problems into concrete visual tasks, making it a fundamental skill in mathematics education.