Problem 70 - 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 \(b\). $$ \begin{array}{ll} A(-0.22,1.6968), & B(-0.12,1.6528) \\ C(1.3,1.028) & D(1.45,0.862) \end{array} $$

Step-by-Step Solution

Answer

The points lie on the line \( y = -0.44x + 1.6 \).

Step 1: Understanding the Equation

We need to determine if the points given lie on a straight line described by the equation \( y = ax + b \). This means the slope \( a \) and the y-intercept \( b \) should be constant for all points.

Step 2: Calculate Slope using Two Points

Choose any two points and calculate the slope \( a \) using the formula \( a = \frac{y_2 - y_1}{x_2 - x_1} \).Let's use points A (-0.22, 1.6968) and B (-0.12, 1.6528):\[a = \frac{1.6528 - 1.6968}{-0.12 + 0.22} = \frac{-0.044}{0.10} = -0.44\]

Step 3: Verify Slope Consistency

We must verify that \( a = -0.44 \) for other pairs of points as well. Let's check with points C (1.3, 1.028) and D (1.45, 0.862):\[a = \frac{0.862 - 1.028}{1.45 - 1.3} = \frac{-0.166}{0.15} = -0.44\]The slope is consistent for these two points.

Step 4: Calculate the Y-intercept

Using the slope \( a = -0.44 \) and one of the points, we calculate \( b \). Using point A (-0.22, 1.6968):\[1.6968 = -0.44(-0.22) + b\]\[b = 1.6968 - 0.0968 = 1.6\]

Step 5: Verify the Equation

We need to check that the equation \( y = -0.44x + 1.6 \) is valid for all points.For B (-0.12, 1.6528):\[1.6528 = -0.44(-0.12) + 1.6 = 1.6528\]For C (1.3, 1.028):\[1.028 = -0.44(1.3) + 1.6 = 1.028\]For D (1.45, 0.862):\[0.862 = -0.44(1.45) + 1.6 = 0.862\]The points satisfy the equation, confirming that they all lie on the same line.

Key Concepts

Slope Calculation

When determining whether multiple points lie on a single straight line, the key component to establish first is the slope. Slope is basically a measure of how steep the line is. To calculate the slope, you can use the formula:

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

The variables \( y_1, y_2 \) represent the y-values of two different points, while \( x_1, x_2 \) represent their corresponding x-values. By plugging in the coordinates of any two points from a dataset, you can determine their slope. For example, using points A (-0.22, 1.6968) and B (-0.12, 1.6528) from our exercise, the calculation is:

• \( a = \frac{1.6528 - 1.6968}{-0.12 + 0.22} = \frac{-0.044}{0.10} = -0.44 \)

This result means that for every unit you move right on the x-axis, the line descends by 0.44 units on the y-axis. Consistency of the slope across different point pairs, as shown in this task, is crucial to confirming they form a straight line.

Y-Intercept Determination

The y-intercept of a line is the exact place where the line crosses the y-axis. This is often denoted by \( b \) in the linear equation \( y = ax + b \). Finding \( b \) requires knowing the slope and using any point on the line.

• The formula is:
  \[ y = ax + b \]

By rearranging, you get:

• \( b = y - ax \)

For instance, using point A (-0.22, 1.6968) and our calculated slope \( a = -0.44 \), we have:
\[ 1.6968 = (-0.44)(-0.22) + b \]Calculating, we find:

• \( b = 1.6968 - 0.0968 = 1.6 \)

This informs us that the line will cross the y-axis at the point \( (0, 1.6) \). Identifying the correct y-intercept is essential as it completes the equation of the line and indicates the line's behavior.

Line Verification

The final step in confirming whether a set of points lies on the same line is to verify the entire line equation with each point. By plugging each point into \( y = ax + b \), you can ascertain whether the equation holds true. If the equation renders accurate y-values for all points, then these points indeed lie on one line.
Start by applying the equation \( y = -0.44x + 1.6 \) to all points from the exercise:

• For B (-0.12, 1.6528):
  \[ 1.6528 = -0.44(-0.12) + 1.6 = 1.6528 \]
• For C (1.3, 1.028):
  \[ 1.028 = -0.44(1.3) + 1.6 = 1.028 \]
• For D (1.45, 0.862):
  \[ 0.862 = -0.44(1.45) + 1.6 = 0.862 \]

Each of these calculations confirms that the given equation accurately describes the relationship among the points. This step is crucial in verifying the solution and ensuring that no calculation errors were made in determining the slope and y-intercept.