Problem 19 - Precalculus Mathematics for Calculus

Question

Find an equation of the line that satisfies the given conditions. Through \((2,3) ;\) slope 5

Step-by-Step Solution

Answer

The equation of the line is \( y = 5x - 7 \).

Step 1: Understand the Point-Slope Form

The point-slope form of a linear equation is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope of the line, and \( (x_1, y_1) \) is a point on the line. In this problem, the slope \( m = 5 \) and the point \( (x_1, y_1) = (2, 3) \).

Step 2: Substitute Given Values

Substitute the given slope \( m = 5 \) and the point \((2, 3)\) into the point-slope form: \( y - 3 = 5(x - 2) \).

Step 3: Simplify the Equation

Distribute the slope \( 5 \) on the right side of the equation: \( y - 3 = 5x - 10 \).

Step 4: Solve for y

Isolate \( y \) by adding 3 to both sides of the equation: \( y = 5x - 7 \). This is the slope-intercept form of the equation of the line.

Key Concepts

Point-Slope Form

The point-slope form is a fantastic way to write an equation of a line when you have a point on the line and the slope. The formula to remember here is \( y - y_1 = m(x - x_1) \).
In this formula:

• \( (x_1, y_1) \) represents a specific point on the line.
• \( m \) is the slope of the line.

This form is extremely useful when you know:

• One point through which the line passes.
• The slope of the line.

You just plug in the values for the slope and the point, and you’ve got the equation of the line. It is a straightforward process and is foundational in understanding how lines work in coordinate geometry. The point-slope form is like a stepping stone to exploring other forms of linear equations, making it essential for solving linear problems quickly.

Linear Equation

A linear equation is an equation that forms a straight line when graphed on a coordinate plane. Its general form is \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. These equations represent relations of the first degree between two variables.
Key properties of linear equations include:

• Simplicity: Only two variables, \( x \) and \( y \), with the degree of each variable being 1.
• Graphing: The solution set of a linear equation forms a straight line.
• Flexibility: Any form of a linear equation can be converted to another.

Linear equations are vital in mathematics as they model a wide range of situations, from financial forecasts to natural phenomena.

Slope-Intercept Form

The slope-intercept form of a linear equation is particularly user-friendly, especially for graphing purposes. The form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept of the line.
Understanding this form involves knowing:

• The slope \( m \) shows how steep the line is and the direction in which it goes.
• The y-intercept \( b \) is where the line crosses the y-axis.

This form offers the quickest insight on how to graph the line since you can start at the y-intercept and use the slope to find other points on the line. Transitioning from point-slope form to slope-intercept form is a common necessity when the goal is to get a clearer picture of the line's behavior on a graph. By isolating \( y \), it's straightforward to visualize and understand how changes in the equation affect the line's position on the coordinate plane.