Problem 45 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Slope \(1,\) through (-7,9)

Step-by-Step Solution

Answer

The equation of the line is \( y = x + 16 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.

Step 2: Identify the Given Values

You are provided with a slope \( m = 1 \) and a point on the line \((-7, 9)\).

Step 3: Substitute the Slope and Point into the Point-Slope Form

The point-slope form of a line equation is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line.Substitute \( m = 1 \), \( x_1 = -7 \), and \( y_1 = 9 \) into the equation:\[y - 9 = 1(x + 7)\]

Step 4: Simplify to Slope-Intercept Form

Distribute the right side of the equation:\[y - 9 = x + 7\]Add 9 to both sides to solve for \( y \):\[y = x + 16\]

Step 5: Write the Final Equation

The equation of the line in slope-intercept form is \( y = x + 16 \). Here, the slope \( m = 1 \), and the y-intercept \( b = 16 \).

Key Concepts

Equation of a Line

An equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of any point on the line. Think of it as a set of instructions telling us where each point on the line is located based on its position along the x-axis.
There are different forms to express this relationship, but one of the most popular is the slope-intercept form. This form is particularly helpful because it gives us both the slope (rate of change, or steepness) and the y-intercept (where the line crosses the y-axis) at once.
The standard slope-intercept form is given by:

• \( y = mx + b \)
• Where \( m \) is the slope and \( b \) is the y-intercept.

By understanding this formula, we can easily sketch the line and predict its path, as long as we know the slope and the y-intercept. This makes analyzing and graphing linear relationships much simpler.

Slope

The slope of a line, often denoted by the letter \( m \), is a measure of the line's steepness or the rate of change between two points on the line. It's like the 'speed' of how much "up" or "down" movement there is per unit of "across" movement along the x-axis.
If you imagine driving on a hill, slope tells you how steep that hill is. In mathematical terms, it's the ratio of the rise (vertical change) to the run (horizontal change) between any two points on the line.
The slope can be calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
• Where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.

A positive slope indicates that the line rises as it moves from left to right, while a negative slope shows it falls. A zero slope means the line is horizontal, indicating no vertical change, and an undefined slope corresponds to a vertical line.

Point-Slope Form

The point-slope form of a linear equation is another useful way to express the equation of a line, especially when you know the slope and a single point on the line. It's particularly handy for constructing a line from incomplete information.
This form is represented as:

• \( y - y_1 = m(x - x_1) \)
• Where \((x_1, y_1)\) is a point on the line, and \( m \) is the slope.

To use the point-slope form, you substitute the slope \( m \) and the coordinates of the given point. As seen in the exercise, the calculation involved substituting the given slope \(1\) and point \((-7, 9)\) into this form.

• This results in \( y - 9 = 1(x + 7) \).

This equation can then be rearranged into the slope-intercept form to find the y-intercept and write the line equation in its most common and user-friendly form.