Equations of Lines and Linear Models: Point-Slope and Slope-Intercept Forms

Equations of Lines and Linear Models

Point-Slope Form

The point-slope form of the equation of a line is a useful way to write the equation of a line when you know the slope and a point on the line. It is especially helpful for constructing equations quickly from given data.

• Definition: The point-slope form of the equation of a line with slope m passing through the point (x1, y1) is:

$y - y_1 = m(x - x_1)$

• Key Points:

  • Use this form when you know a point and the slope.

  • Can be rearranged to other forms, such as slope-intercept or standard form.

• Example: Write the equation of the line through (-4, 1) with slope -3.

  • Given: $x_1 = -4$, $y_1 = 1$, $m = -3$

  • Substitute into the formula:

  $y - 1 = -3(x + 4)$

• Practice: Write the equation of the line through (-5, 4) with slope -\frac{1}{2}.

  • Given: $x_1 = -5$, $y_1 = 4$, $m = -\frac{1}{2}$

  • Substitute:

  $y - 4 = -\frac{1}{2}(x + 5)$

Point-Slope Form Using Two Points

When two points are given, you can first find the slope and then use the point-slope form.

• Finding the Slope: For points $(x_1, y_1)$ and $(x_2, y_2)$, the slope is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: Find the equation of the line through $(-3, 2)$ and $(2, 4)$.

  • Calculate slope: $m = \frac{4 - 2}{2 - (-3)} = \frac{2}{5}$

  • Use point-slope form with $(-3, 2)$:

  $y - 2 = \frac{2}{5}(x + 3)$

• Standard Form: Rearranging to $Ax + By = C$:

  • Multiply both sides by 5: $5(y - 2) = 2(x + 3)$

  • Expand: $5y - 10 = 2x + 6$

  • Rearrange: $-2x + 5y = 16$ or $2x - 5y = -16$

Slope-Intercept Form

The slope-intercept form is a widely used way to express the equation of a line, especially for graphing and identifying the slope and y-intercept directly.

• Definition: The slope-intercept form is:

$y = mx + b$

• Where m is the slope and b is the y-intercept (the value of y when x = 0).

• Converting to Slope-Intercept Form: Solve for y in any linear equation.

• Example: Convert $4x + 5y = 10$ to slope-intercept form.

  • Isolate y: $5y = -4x + 10$

  • Divide by 5: $y = -\frac{4}{5}x + 2$

• Practice: Convert $x + 2y = -4$ to slope-intercept form.

  • Isolate y: $2y = -x - 4$

  • Divide by 2: $y = -\frac{1}{2}x - 2$

Using Slope-Intercept Form with Two Points

Given two points, you can find the slope, then use one point to solve for the y-intercept.

• Example: Find the equation of the line through $(2, 3)$ and $(-1, 2)$.

  • Find the slope: $m = \frac{3 - 2}{2 - (-1)} = \frac{1}{3}$

  • Use point $(2, 3)$ in $y = mx + b$:

  • $3 = \frac{1}{3}(2) + b \implies b = 3 - \frac{2}{3} = \frac{7}{3}$

  • Equation: $y = \frac{1}{3}x + \frac{7}{3}$

Graphing Linear Equations

To graph a line using its equation, identify the slope and y-intercept from the slope-intercept form, or use two points to plot the line.

• Steps:

  1. Plot the y-intercept $(0, b)$.

  2. Use the slope $m = \frac{rise}{run}$ to find another point.

  3. Draw a straight line through the points.

• Example: For $y = -\frac{1}{3}x - 1$, the slope is $-\frac{1}{3}$ and the y-intercept is $-1$.

Finding an Equation from a Graph

You can determine the equation of a line by identifying the slope and y-intercept from its graph.

• Steps:

  1. Find two points on the line.

  2. Calculate the slope: $m = \frac{rise}{run}$.

  3. Identify the y-intercept (where the line crosses the y-axis).

  4. Write the equation in $y = mx + b$ form.

• Example: If the slope is $-2$ and the y-intercept is $1$, the equation is$y = -2x + 1$.

Summary Table: Forms of Linear Equations

Form	Equation	When to Use
Point-Slope	$y - y_1 = m(x - x_1)$	Given a point and the slope
Slope-Intercept	$y = mx + b$	Given the slope and y-intercept, or for graphing
Standard	$Ax + By = C$	For certain algebraic manipulations or applications

Additional info: These notes cover the core concepts of writing equations of lines in various forms, finding equations from points or graphs, and graphing linear equations, which are foundational skills in College Algebra.