Problem 48 - Algebra 1

Question

Write an equation in standard form of the line that passes through the two points. $$(9,-2),(-3,2)$$

Step-by-Step Solution

Answer

The equation of the line in standard form is \( x + 3y - 3 = 0 \).

Step 1: Determine the Slope

The first step involves finding the slope of the line using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the coordinates of the two points into the equation, we get \( m = \frac{2 - (-2)}{-3 - 9} = -\frac{1}{3} \).

Step 2: Obtain the Equation in Slope-Intercept Form

With the slope, we can use the slope-point formula to get the equation in slope-intercept form which is \( y - y_1 = m(x - x_1) \). Plugging in the slope and the coordinates of the first point we get \( y - (-2) = -\frac{1}{3}(x - 9) \) which simplifies to \( y = -\frac{1}{3}x + 1 \).

Step 3: Convert to Standard Form

The final step is to convert the equation from slope-intercept form to standard form by aligning it with the 'Ax + By = C' format. By moving all terms to one side of the equation, we get \( \frac{1}{3}x + y - 1 = 0 \). However, to avoid the fraction, we can multiply every term by 3 to get \( x + 3y - 3 = 0 \) which satisfies requirements for the standard form.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when dealing with linear equations. The slope measures the steepness and direction of a line and is usually represented by the letter 'm'.
To find the slope between two points, \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the formula used is \(\frac{y_2 - y_1}{x_2 - x_1}\). This formula subtracts the y-coordinate of point A from that of point B and divides the result by the subtraction of the x-coordinate of point A from that of point B.
If the slope is positive, the line rises as it moves from left to right. Conversely, if the slope is negative, the line falls. A slope of zero means the line is horizontal, and an undefined slope (where \( x_2 - x_1 = 0 \)) means the line is vertical.

Slope-Intercept Form

The slope-intercept form is an intuitive way of writing the equation of a line. It is expressed as \( y = mx + b \), where 'm' is the slope, and 'b' is the y-intercept—the point where the line crosses the y-axis.
Starting with a slope and one point on the line, the slope-intercept form can easily be constructed. For the equation given in the exercise, the slope \( -\frac{1}{3} \) is paired with the y-intercept (0,1) to form \( y = -\frac{1}{3}x + 1 \).

Advantages of Slope-Intercept Form

• Immediate visualization of the line's slope and y-intercept.
• Convenient for graphing the line.
• Useful for comparing two lines to check for parallel or perpendicular characteristics.

Point-Slope Form

Point-slope form is ideal for when you know a point on a line and its slope. This form is described by the equation \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line.
In our example, by knowing the slope and choosing one of the points, say \( (9, -2) \), we can plug these values in to get \( y - (-2) = -\frac{1}{3}(x - 9) \). This form is particularly useful because it showcases how the vertical change of the line (\( y - y_1 \)) relates to its horizontal change (\( x - x_1 \)).

When to Use Point-Slope Form

• When a specific point and the slope are known.
• Ideal for writing equations quickly without needing to solve for y-intercept.
• Useful for understanding the relative position of a line to a specific point.