Problem 22 - Technical Mathematics with Calculus

Question

Write the equation of each straight line passing through the given points and make a graph. $$(3.22,2.53) \text { and }(3.51,-2.54)$$

Step-by-Step Solution

Answer

The equation of the line is \(y = -157.97(x - 3.22) + 2.53\) after simplifying, and the graph is a straight line passing through the points \((3.22,2.53)\) and \((3.51,-2.54)\).

Step 1: Determine the Slope

The slope of a line (m) passing through two points \text{Point 1} (x_1, y_1) and \text{Point 2} (x_2, y_2) is determined by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For the given points (3.22, 2.53) and (3.51, -2.54), the slope is calculated as \(m = \frac{-2.54 - 2.53}{3.51 - 3.22}\).

Step 2: Calculate the Slope

Substitute the coordinates of the points into the slope formula: \(m = \frac{-2.54 - 2.53}{3.51 - 3.22}\), then simplify to find the value of the slope.

Step 3: Use the Point-Slope Formula

Once the slope is found, use the point-slope formula \(y - y_1 = m(x - x_1)\) to write the equation of the line. Use one of the given points for \(x_1\) and \(y_1\). Let's use the first point (3.22, 2.53).

Step 4: Write the Equation of the Line

Substitute the slope we found in Step 2 and the coordinates of the point (3.22, 2.53) into the point-slope formula and simplify to get the equation of the line in slope-intercept form, which is \(y = mx + b\).

Step 5: Graph the Equation

Plot the two given points on a coordinate plane. Draw a line through them, and extend it in both directions. Make sure the slope of the line matches the slope calculated in Step 2.

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line on a coordinate plane is essential for graphing and working with linear equations. Slope, often represented as m, is a measure of the steepness and the direction of a line. It is calculated using the difference in the y-coordinates (rise) over the difference in the x-coordinates (run) between two distinct points on the line. Mathematically, it is expressed as

\( m = \frac{y_2 - y_1}{x_2 - x_1} \)

To compute the slope with this formula, you simply need to subtract the y-coordinate of the first point from the y-coordinate of the second point and divide that by the subtraction of the x-coordinate of the first point from the x-coordinate of the second. For example, if we have the points (3.22, 2.53) and (3.51, -2.54), the slope m will be calculated as follows:

\( m = \frac{-2.54 - 2.53}{3.51 - 3.22} \)

The resulting value tells us how many units the line rises (or falls) for each unit it runs horizontally. Remember, a positive slope indicates the line rises from left to right, while a negative slope indicates the line falls from left to right.

Point-Slope Formula

Once the slope of a line is determined, the point-slope formula becomes a handy tool to write the equation of the line. This formula is especially useful when you have the slope and any point that the line passes through. The point-slope formula is written as

\( y - y_1 = m(x - x_1) \)

Where (x_1, y_1) is the given point on the line and m is the slope. To use this formula, you replace m with the slope you calculated and (x_1, y_1) with the coordinates of one of your given points. For instance, using the point (3.22, 2.53) from our previous example and the slope we calculated, the equation would be set up as:

\( y - 2.53 = m(x - 3.22) \)

After substituting the value of the slope m, you can then rearrange the equation into the slope-intercept form, which is typically y = mx + b.

Graphing Linear Equations

Graphing linear equations involves plotting points, lines, and understanding the relationship between the algebraic equation and its graphical representation. When we have an equation of a line, we use it to plot points on the coordinate plane that satisfies the equation. To graph the line, you can start by plotting the points you already know—in this case, the points (3.22, 2.53) and (3.51, -2.54). After marking these on the graph, the next step is to draw a line through them. To ensure accuracy, use additional points that satisfy the equation, or simply extend the line in both directions making sure that it aligns with the slope of the line calculated earlier. This visual representation is a powerful tool in understanding the behavior of the line, its direction, and where it may intersect with the axes.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and curves. It consists of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Their intersection is the origin, labeled as the point (0,0).

Every point on the plane is defined by an ordered pair of numbers (x, y), where x represents the horizontal position relative to the origin, and y represents the vertical position. In graphing, the coordinate plane allows us to represent algebraic equations visually, and it serves as a fundamental tool in various fields of mathematics and science. It's essential to be comfortable navigating this plane when working with any form of graphing problem.