Problem 5 - College Algebra with Modeling and Visualization

Question

Find a point-slope form of the line satisfying the conditions. Use the first point given for \(\left(x_{1}, y_{1}\right) .\) Then convert the equation to slope-intercept form. Slope \(-2.4,\) passing through \((4,5)\)

Step-by-Step Solution

Answer

Point-slope form: \( y - 5 = -2.4(x - 4) \); Slope-intercept form: \( y = -2.4x + 14.6 \).

Step 1: Identify the Variables

The point-slope form of a line is given by the formula \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope, and \( (x_1, y_1) \) is a point on the line. We are given the slope \( m = -2.4 \) and the point \( (4, 5) \). So, we have \( x_1 = 4 \) and \( y_1 = 5 \).

Step 2: Substitute into Point-Slope Formula

Substitute the given values into the point-slope form equation: \[ y - 5 = -2.4(x - 4) \].

Step 3: Distribute the Slope

Apply the distributive property to \( -2.4(x - 4) \): \[ y - 5 = -2.4x + 9.6 \].

Step 4: Solve for y (Slope-Intercept Form)

To convert to slope-intercept form \( y = mx + b \), add 5 to both sides of the equation: \[ y = -2.4x + 9.6 + 5 \]. Simplify the equation: \[ y = -2.4x + 14.6 \].

Step 5: Final Slope-Intercept Form

The final slope-intercept form of the equation is \( y = -2.4x + 14.6 \).

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line in two dimensions. It's written as \( y = mx + b \) where:

• \( y \) represents the dependent variable or output of the equation.
• \( m \) is the slope of the line, showing the rate of change of \( y \) with respect to \( x \).
• \( x \) is the independent variable or input of the equation.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis (when \( x = 0 \)).

The slope-intercept form is particularly useful because it immediately gives us both the direction and the steepness of the line with the slope \( m \), and the position of the line with the y-intercept \( b \). For example, having the equation \( y = -2.4x + 14.6 \), it tells us that for every increase in \( x \) by 1 unit, \( y \) will decrease by 2.4 units, starting at 14.6 on the y-axis.

distributive property

The distributive property is a key algebraic principle used to simplify expressions and equations. It states that a term multiplied by a group of terms inside a parenthesis can be distributed, or multiplied, to each term within the parenthesis separately. For instance, when you have an expression like \( a(b + c) \), it can be simplified or expanded to \( ab + ac \).This property is essential when converting equations, like going from point-slope form to slope-intercept form. In our example, we had the expression \(-2.4(x - 4)\). Using the distributive property allows us to expand that to \(-2.4x + 9.6 \), effectively spreading the multiplication over both \( x \) and \(-4 \) within the parenthesis. This step is crucial before you can isolate the \( y \) variable and express the line in slope-intercept form. Always remember, the distributive property helps simplify and rearrange equations, making them easier to interpret and solve.

linear equations

Linear equations represent straight lines on a graph and are fundamental in understanding algebra and geometry. They are equations of the first order, meaning they involve no higher powers or roots of the variables. Generally expressed in the form \( ax + by = c \) or, more familiarly, \( y = mx + b \), these equations define the relationship between \( x \) and \( y \) in a linear manner.Characteristics of linear equations include:

• Constant slope: The slope \( m \) remains the same throughout the line.
• Linear relationship: A uniform rate of change, meaning there's a constant increment or decrement for every unit change in \( x \).
• Graph: Plots as a straight line on a coordinate plane, where each point on the line satisfies the equation.

In the given exercise, we worked with a linear equation in the form \( y = -2.4x + 14.6 \). Here, the equation describes a line where for every one unit increase in \( x \), \( y \) decreases by 2.4 units, demonstrating the linear relationship between \( x \) and \( y \). Linear equations are used in numerous disciplines, making them a crucial part of mathematical learning.