Problem 69 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 69-70: The given points were found using empirical methods. Determine whether they lie on the same line \(y=a x+b\), and if so, find the values of \(a\) and \(b\). $$ \begin{array}{ll} A(-1.3,-1.3598), & B(-0.55,-1.11905) \\ C(1.2,-0.5573), & D(3.25,0.10075) \end{array} $$

Step-by-Step Solution

Answer

All points lie on the line \(y = 0.321x - 0.9425\).

Step 1: Understanding the Line Equation

The equation of a line in slope-intercept form is given by \(y = ax + b\), where \(a\) is the slope and \(b\) is the y-intercept. To determine if all points lie on the same line, each point \((x_i, y_i)\) should satisfy the equation \(y_i = ax_i + b\).

Step 2: Calculate the Slope (a) using Two Points

Choose any two points to calculate the slope \(a\). Using points \(A(-1.3, -1.3598)\) and \(B(-0.55, -1.11905)\), calculate the slope as follows: \[ a = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1.11905 - (-1.3598)}{-0.55 - (-1.3)} = \frac{0.24075}{0.75} = 0.321 \]

Step 3: Calculate Y-Intercept (b) with One Point

Use the slope \(a = 0.321\) and any one point to find the y-intercept \(b\). Using point \(A(-1.3, -1.3598)\):\[ -1.3598 = 0.321(-1.3) + b \]Solve for \(b\):\[ -1.3598 = -0.4173 + b \]\[ b = -1.3598 + 0.4173 \]\[ b = -0.9425 \]

Step 4: Verify Line Equation with Other Points

Use the line equation \(y = 0.321x - 0.9425\) and check if it passes through points \(C(1.2, -0.5573)\) and \(D(3.25, 0.10075)\).For point \(C\):\[ y = 0.321(1.2) - 0.9425 = 0.3852 - 0.9425 = -0.5573 \] which is correct.For point \(D\):\[ y = 0.321(3.25) - 0.9425 = 1.04325 - 0.9425= 0.10075 \] which is correct.

Step 5: Conclusion

Since all points satisfy the line equation \(y = 0.321x - 0.9425\), they all lie on the same line. The values of \(a\) and \(b\) are 0.321 and -0.9425, respectively.

Key Concepts

Slope-Intercept Form

When working with linear equations, a fundamental concept is the slope-intercept form. This is typically written as \(y = ax + b\), where \(a\) represents the slope of the line, and \(b\) signifies the y-intercept. To understand why this form is so useful, think of it as a template that allows you to quickly sketch a line by knowing just two key pieces of information: how steep the line is (the slope) and where it crosses the y-axis (the y-intercept).

One major advantage of using the slope-intercept form is its simplicity and ease of use in various applications. If you can determine the slope and y-intercept for a set of data points, you can easily form the equation of a line that represents the relationship between this data. This format is also highly effective for graphing purposes, making it straightforward to predict how changes in \(x\) affect \(y\), which is critical in both educational and real-world contexts.

Slope Calculation

Calculating the slope is a crucial step in understanding the nature of a linear relationship. The slope of a line is a measure of its steepness or incline, typically denoted by \(a\) in the linear equation \(y = ax + b\). You calculate the slope by taking any two points on the line, designated here as \((x_1, y_1)\) and \((x_2, y_2)\).

The formula for calculating the slope \(a\) is:

• \(a = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula essentially tells you how much \(y\) increases or decreases for a unit change in \(x\).

For example, when you used points \((-1.3, -1.3598)\) and \(-0.55, -1.11905)\), the slope \(a\) was calculated to be \(0.321\). This indicates that for every one unit increase in \(x\), \(y\) increases by \(0.321\) units. Understanding how to calculate and interpret the slope is fundamental, as it shows the rate of change and direction of the line.

Y-Intercept Calculation

Once you have determined the slope, the next step is to calculate the y-intercept, \(b\). The y-intercept of a line is where it crosses the y-axis when \(x\) is zero. This point provides a starting value for the line when graphing.

To find the y-intercept, you use the slope \(a\) and choose one of the known points on the line. Using the formula from the line equation \(y = ax + b\), you can isolate \(b\) by substituting \(x\), \(y\), and \(a\) into the equation:

• \(y = ax + b\)
• (solve for \(b\)) \(b = y - ax\)

With the point \((-1.3, -1.3598)\) and the slope \(0.321\), the calculation was made as follows:

• \(-1.3598 = 0.321(-1.3) + b\)
• \(b = -1.3598 + 0.4173\)
• \(b = -0.9425\)

The y-intercept of \(-0.9425\) indicates that when \(x = 0\), \(y\) is \(-0.9425\). This information helps in fully describing the line and accurately graphing it.