Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the general form of the equation of the line that passes through the two points. \((3,-3),(10,-1)\)

Step-by-Step Solution

Answer

The general form of the equation of the line passing through the points \((3, -3)\) and \((10, -1)\) is \(2x - 7y = 17\).

Step 1: Find the slope

Calculate the slope \(m\) of the line by plugging the coordinates of the two points into the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). So, \(m=\frac{-1 - (-3)}{10 - 3} = \frac{2}{7}\)

Step 2: Find the y-intercept

Substitute one of the given points (for example, \((3, -3)\)) and the slope \(m\) into the point-slope form \(y = mx + c\) to solve for \(c\). From the equation -3 = \(\frac{2}{7}\) * 3 + \(c\), solve for \(c\) to get \(c = -\frac{17}{7}\)

Step 3: Find the general form

The equation can now be re-written in general form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers and \(A > 0\). Multiply the entire equation by 7 to get integers and avoid fractions: 7y = 2x - 17. Re-arrange to general form: \(2x - 7y = 17\).

Key Concepts

Slope Calculation

When establishing the equation of a line, the first critical step is calculating the slope, denoted by \( m \). The slope illustrates how steep the line is, indicating the rate of change between the two variables along the line. To determine the slope between two points, \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, subtract the first y-coordinate from the second y-coordinate, then subtract the first x-coordinate from the second x-coordinate.

• For instance, with the points \((3, -3)\) and \((10, -1)\), you get: \( m = \frac{-1 - (-3)}{10 - 3} = \frac{2}{7} \).
• The slope \( m = \frac{2}{7} \) suggests that for each increase of 7 units horizontally, the line increases by 2 units vertically.

Understand that slope is essential as it defines how "tilted" the line is, helping in predicting the line's behavior without plotting it.

Point-Slope Form

Once you have calculated the slope \( m \), the next step is using it in the point-slope form of a line equation. The point-slope form is particularly useful when you know a point on the line and its slope. The formula looks like this: \[ y - y_1 = m(x - x_1) \] Here \((x_1, y_1)\) is a known point, and \( m \) is the slope.

• For our points \((3, -3)\), this becomes \( y + 3 = \frac{2}{7}(x - 3) \).

Rearrange the formula if necessary to plug in your known point properly. This form highlights that you can immediately start at a known point and "move" along the slope to understand the line's path.

General Form of a Line

The grand showcase of linear equations often leads you to the general form. A linear equation in general form follows: \( Ax + By = C \). This format is comprehensive and can be easily interpreted or converted into other forms. Let's see how to shift from point-slope to the general form:

• Start by rearranging the point-slope equation: \( y + 3 = \frac{2}{7}(x - 3) \) yields \( y = \frac{2}{7}x - \frac{6}{7} - 3 \).
• Multiply all terms by 7 to eliminate fractions: \( 7y = 2x - 17 \).
• Reorganize to get \( 2x - 7y = 17 \).

This conversion ensures all coefficients are integers, and \( A \) is positive. The equation shows a classic standard that is simple, neat, and especially useful for solving simultaneous equations.

Y-Intercept Calculation

In linear equations, finding the y-intercept is about determining where the line crosses the y-axis. It's represented by \( c \) in the equation \( y = mx + c \). To find this value:

• Use the slope-intercept form. Plug in known values to solve for \( c \).
• For point \((3, -3)\) and slope \( \frac{2}{7} \), substitute to get \( -3 = \frac{2}{7} \times 3 + c \).
• This simplifies to \( c = -\frac{17}{7} \).

Understanding \( c \) is pivotal for graphing as it pinpoints where exactly the line begins on the y-axis, forming a base for sketching and interpreting the linear function.