Problem 43 - Introductory Algebra for College Students

Question

I use \(y=m x+b\) to write equations of lines passing through two points when neither contains the \(y\) -intercept.

Step-by-Step Solution

Answer

The equation of the line is \(y = mx + b\) where \(m\) is the slope which can be calculated using any two points on the line and \(b\) is the y-intercept which can be found by substituting any point on the line and calculated slope into the line equation.

Step 1: Calculate the Slope

The slope \(m\) of the line is calculated by taking two different points (x1, y1) and (x2, y2) on the line, and using the formula \[ m = \frac {y2 - y1} {x2 - x1} \]

Step 2: Substitute a Point into the Line Equation to Solve for the Y-intercept 'b'

Once the slope \(m\) is determined, choose either of the two given points and substitute it into the line equation \(y = mx + b\) along with the slope, and solve for \(b\), the y-intercept. For a given point (x1, y1), the formula will look like this \[y1 = m*x1 + b\]. After substituting \(m\), \(x1\) and \(y1\) in this formula, solve for \(b\).

Step 3: Write the Equation of the Line

Now, we can write down the equation of the line with known slope \(m\) and y-intercept \(b\) in the general form \(y = mx + b\).

Key Concepts

Slope Calculation

The first step in finding the equation of a line when given two distinct points is to determine the slope. The slope is essentially a number that describes both the direction and the steepness of the line. We represent this in mathematics with the letter 'm'.
To calculate the slope, use the slope formula: \[ m = \frac {y_2 - y_1} {x_2 - x_1} \].
This involves using two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\). The difference in the 'y' values of the two points, labeled as \(y_2 - y_1\), represents the vertical change, while \(x_2 - x_1\), the difference in the 'x' values, represents the horizontal change.

• If the slope, 'm', is positive, the line rises from left to right.
• If 'm' is negative, the line falls from left to right.
• If 'm' is zero, the line is horizontal.
• An undefined slope typically means a vertical line.

By calculating the slope, we get the first crucial piece of information about the line.

Y-intercept

Once the slope 'm' is calculated, the next task is to find the 'y-intercept'. The y-intercept is the point where the line crosses the y-axis. In simpler terms, it is the y-value when the x-value is zero. We represent this with the letter 'b' in the line equation, \(y = mx + b\).
To find 'b', substitute one of the given points and the calculated slope into the equation and solve for 'b'. For instance, if you have the point \((x_1, y_1)\), substituting it into the equation gives us \(y_1 = mx_1 + b\).
Solving this equation helps to determine 'b'.

• ensure you substitute carefully to avoid errors, this step solidifies the exact position of the line on the graph.
• Knowing the y-intercept allows you to graph the line precisely.

By finding the y-intercept, you now have the second essential component to completely describe the line.

Two-Point Form

The two-point form is especially useful when you have two points but don't have prior knowledge of the line equation or y-intercept. This approach allows you to derive the equation of a line by using these points directly, without needing additional information.
The process involves two steps: calculating the slope with the two-point formula and then using the slope-intercept formula to solve for the line's equation. When using two-point form, you essentially manipulate the formula to express the line in a more functional form: \(y = mx + b\).

• Use the two given points to find the slope, as detailed in the 'Slope Calculation' section.
• Use any of the points to find the y-intercept, following the instructions in 'Y-intercept'.
• Combine these to establish an equation in the format \(y = mx + b\).

Thus, using the two-point form, you can efficiently bridge the gap from having sparse data points to crafting a full line equation.