Problem 55 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Slope \(-\frac{4}{7},\) through (-1,-2)

Step-by-Step Solution

Answer

y = -\frac{4}{7}x - \frac{18}{7}

Step 1: Understand the Slope-Intercept Form

The equation of a line in slope-intercept form is given by \( y = mx + c \) where \( m \) is the slope and \( c \) is the y-intercept. We have a slope \( m = -\frac{4}{7} \).

Step 2: Use the Point-Slope Formula

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. Use the given point \( (-1, -2) \) and slope \( -\frac{4}{7} \): \( y - (-2) = -\frac{4}{7}(x - (-1)) \).

Step 3: Simplify the Equation

Simplify \( y + 2 = -\frac{4}{7}(x + 1) \). Distribute \( -\frac{4}{7} \): \( y + 2 = -\frac{4}{7}x - \frac{4}{7} \).

Step 4: Solve for y in Slope-Intercept Form

Isolate \( y \) by subtracting \( 2 \) from both sides: \( y = -\frac{4}{7}x - \frac{4}{7} - 2 \). Convert \( 2 \) to \( \frac{14}{7} \): \( y = -\frac{4}{7}x - \frac{4}{7} - \frac{14}{7} \). Combine the constants: \( y = -\frac{4}{7}x - \frac{18}{7} \).

Key Concepts

Equation of a Line

An equation of a line represents a straight path that continues infinitely in both directions on a graph. To form this equation, it's important to know a point on the line and the slope, or how steep the line is. Lines can be expressed in multiple forms, but a common one is the slope-intercept form, which we'll explore further.

• The equation of a line conveys its slope and intercept;
• These equations can be rearranged based on what information is given;
• Understanding different forms allows for flexibility when solving or graphing equations.

Recognizing the parts of the equation and how they interact is key to mastering linear equations in analytics and graphing settings.

Point-Slope Form

The point-slope form is another way to write an equation of a line, especially useful when a point on the line and the slope are known. Unlike the slope-intercept form, which highlights the y-intercept, this form uses:

• A known point \( (x_1, y_1) \) on the line;
• The slope, \( m \), of the line.

The equation is expressed as \( y - y_1 = m(x - x_1) \). This form is ideal for quickly forming an equation with inputted values. For example, using the point \( (-1, -2) \) and slope \( -\frac{4}{7} \), we can substitute these values directly into the formula, making this form convenient for transformations into other formats like slope-intercept.

Slope and Intercept

Understanding slope and intercept is fundamental when working with linear equations. The slope \( m \) measures the steepness of a line, indicating how much the y-value changes with each step along the x-axis.

• The slope is positive if the line rises as it moves right;
• It's negative if the line falls as it moves right.

The intercept, on the other hand, is where the line crosses the y-axis, noted by the 'c' in the slope-intercept form \( y = mx + c \). Thus, every line can be defined by these two characteristics. In our exercise, the line's slope \( -\frac{4}{7} \) tells us it decreases as x increases, showing it's slanted downward with the y-intercept at \( -\frac{18}{7} \).

Linear Equations

Linear equations form the backbone of many mathematical concepts, representing constant rates of change and models of proportionality. These equations, like the one \( y = -\frac{4}{7}x - \frac{18}{7} \), produce straight lines when graphed and analyze the relationship between two variables.

• They follow the pattern of a first-degree polynomial;
• All their points align perfectly along one straight line.

The simplicity of linear equations makes them incredibly useful not only in mathematics but also in physics, economics, and engineering to illustrate trends and make predictions based on data sets. The ease of transforming one form to another only adds to their versatility and application.