Problem 20 - A Graphical Approach to College Algebra

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((-1,6.25)\) and \((2,-4.25)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -3.5x + 2.75 \).

Step 1: Identify Points

We have two points on the line: \((-1, 6.25)\) and \((2, -4.25)\). These points are denoted as \((x_1, y_1) = (-1, 6.25)\) and \((x_2, y_2) = (2, -4.25)\).

Step 2: Calculate Slope

The slope \(m\) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the given points into the formula:\[ m = \frac{-4.25 - 6.25}{2 - (-1)} = \frac{-10.5}{3} = -3.5 \]

Step 3: Use Slope-Intercept Form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We found \(m = -3.5\). We now choose one point to solve for \(b\). Let's use \((-1, 6.25)\).

Step 4: Solve for Y-Intercept

Substitute \(m = -3.5\), \(x = -1\), and \(y = 6.25\) into the equation \(y = mx + b\):\[ 6.25 = -3.5(-1) + b \]\[ 6.25 = 3.5 + b \]Subtract \(3.5\) from both sides to solve for \(b\):\[ b = 6.25 - 3.5 = 2.75 \]

Step 5: Write the Final Equation

Insert the values for \(m\) and \(b\) into the slope-intercept form:\[ y = -3.5x + 2.75 \]This is the equation of the line in slope-intercept form.

Key Concepts

Equation of a Line

When we talk about an equation of a line, we're dealing with a rule that represents a straight path in a coordinate plane. One common way to express this is through the slope-intercept form, which is written as:

• \( y = mx + b \)

In this equation:

• \( y \) is the dependent variable, which changes with \( x \)
• \( x \) is the independent variable
• \( m \) represents the slope of the line
• \( b \) is the y-intercept, or where the line crosses the y-axis

This form is particularly useful because it allows you to quickly identify the slope and y-intercept of a line, making it easier to graph or analyze. A big part of understanding lines is learning how to express them using different forms, with the slope-intercept form being one of the most straightforward and widely used.

Slope Calculation

The slope of a line is a crucial concept as it indicates the line's steepness and direction. It's calculated by comparing the vertical change (rise) to the horizontal change (run) between two points on a line. The formula for slope \( m \) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In our original problem, the points given were

• \((-1, 6.25)\) and \((2, -4.25)\)

By substituting these into the slope formula, we find:

• \( m = \frac{-4.25 - 6.25}{2 - (-1)} = \frac{-10.5}{3} = -3.5 \)

This slope tells us that for every 1 unit increase in \( x \), the \( y \) value decreases by 3.5 units. A negative slope like \(-3.5\) shows the line is going downwards as you move from left to right.

Y-Intercept

The y-intercept is another key feature of a line, representing the point where the line crosses the y-axis. In the context of the slope-intercept equation \( y = mx + b \), the value of \( b \) is the y-intercept.

• Finding \( b \) involves rearranging the slope-intercept equation so that when \( x = 0 \), the value of \( y \) is \( b \).

In the step-by-step solution, the y-intercept \( b \) was found by substituting the slope \( m = -3.5 \) and one of the given points, say

• \((-1, 6.25)\)

into the equation:

• \( 6.25 = -3.5(-1) + b \) leads us to find \( b = 2.75 \)

This calculation reveals that the line will cross the y-axis at \( 2.75 \), indicating the initial value of \( y \) when \( x \) is zero. Grasping how slopes and intercepts shape lines helps in predicting and understanding linear relationships.