Problem 12 - Technical Mathematics with Calculus

Question

Find the slope and the \(y\) intercept for each equation, and make a graph. $$y=7 x+2$$

Step-by-Step Solution

Answer

The slope of the equation is 7, and the y-intercept is 2. To graph, plot the y-intercept at (0, 2) and use the slope to find another point, then draw the line through these points.

Step 1: Identify the Slope

The equation is in the slope-intercept form, which is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The coefficient of \(x\) in the given equation is 7, which is the slope: \(m = 7\).

Step 2: Identify the Y-Intercept

The constant term in the equation \(y=7x+2\) represents the y-intercept, which is the point where the line crosses the y-axis. The y-intercept is 2, given by the term \(+2\).

Step 3: Plot the Y-Intercept

On a graph, plot the y-intercept \((0, 2)\). This point is where the line will cross the y-axis.

Step 4: Use the Slope to Find Another Point

Starting from the y-intercept, use the slope to find another point on the line. Since the slope is 7, which means \(\frac{rise}{run} = \frac{7}{1}\), from the y-intercept, move up 7 units (rise) and 1 unit to the right (run). Plot this second point.

Step 5: Draw the Line

Draw a straight line through the two points. This is the graph of the equation \(y=7x+2\).

Key Concepts

Understanding the Slope of a Line

The slope of a line is a measure of its steepness and direction. In the context of the slope-intercept form of a linear equation, like the equation \(y = 7x + 2\), the slope is represented by the coefficient of \(x\), which in this case is 7. This number tells us that for each unit we move to the right on the \(x\)-axis, the value of \(y\) increases by 7 units, illustrating a steep upward trend.

The slope can be thought of as the ratio of the 'rise' over the 'run' between any two points on the line. If we pick two points on the graph and draw a right triangle between them, the vertical side (rise) represents the change in \(y\), while the horizontal side (run) represents the change in \(x\). A positive slope, as seen in \(y = 7x + 2\), indicates that the line is rising from left to right, while a negative slope would imply a fall.

Identifying the Y-Intercept

The y-intercept is the point where the graph of a line crosses the y-axis, indicating the value of \(y\) when \(x = 0\). To find the y-intercept from an equation in slope-intercept form, you look for the constant term. In our example equation \(y = 7x + 2\), the y-intercept is 2. This means that the graph of this line will pass through the point \(0, 2\) on the y-axis.

To visualize this on the coordinate plane, you can simply plot the point \(0, 2\) directly onto the y-axis. This will always be the first point you plot when graphing a linear equation in slope-intercept form. From there, the slope guides you to additional points for drawing the actual line.

Graphing Linear Equations

Graphing linear equations is a step-by-step process that visually represents the relationship between two variables. After determining the slope and y-intercept, the equation can easily be graphed on a coordinate plane. For the equation \(y = 7x + 2\), start by plotting the y-intercept \(0, 2\).

Then, apply the slope: since the slope is 7, from the y-intercept, go up 7 units and 1 unit to the right to find the next point on the graph. This might be challenging due to the steepness, but it's accurate according to the slope value. After plotting the second point, simply draw a straight line through both points to complete the graph. The line extends infinitely in both directions, but two points are sufficient to establish the line's path on the graph.