Problem 71 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ m=-2, \quad(0,2) $$

Step-by-Step Solution

Answer

Answer: The equation of the line in slope-intercept form is y = -2x + 2.

Step 1: Write the general equation of the line in slope-intercept form

The general equation of the line in slope-intercept form is: $$ y = mx + b $$

Step 2: Substitute the given slope and point into the equation

We are given the slope \(m=-2\) and the point \((x,y)=(0,2)\). Substitute this information into the equation: $$ 2 = (-2)(0) + b $$

Step 3: Solve for the y-intercept, b

The equation becomes: $$ 2 = 0 + b $$ Therefore, the y-intercept \(b\) is: $$ b = 2 $$

Step 4: Write the equation of the line using the slope and y-intercept

Now that we have both the slope \(m=-2\) and the y-intercept \(b=2\), we can write the equation of the line in slope-intercept form as: $$ y = -2x + 2 $$

Key Concepts

Linear Equations

Linear equations represent straight lines when graphed on a coordinate plane. These equations come in various forms, such as standard form and point-slope form, but the slope-intercept form is particularly popular because it clearly shows the slope and y-intercept. An equation in slope-intercept form is written as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.
The unique feature of linear equations is that they have a constant rate of change. This means that for every increase or decrease in \( x \), the \( y \) value changes by a consistent amount, determined by the slope \( m \).
To create a linear equation, you need to know two pieces of information: the slope \( m \) and the y-intercept \( b \). With these values, you can plot the line or predict \( y \) for any \( x \) value by simply plugging \( x \) into the equation.

• The slope \( m \) tells you how the line rises or falls.
• The y-intercept \( b \) shows where the line crosses the y-axis.

Understanding linear equations is essential because they serve as a foundation for more complex algebraic concepts.

y-intercept

The y-intercept of a linear equation is the point where the line crosses the y-axis. It is represented by \( b \) in the slope-intercept form equation \( y = mx + b \). This point is crucial because it provides a starting point for graphing the line.
To find the y-intercept, you can set \( x = 0 \) in the equation. In this exercise, the point given is \((0,2)\), directly showing that the line crosses the y-axis at 2, hence \( b = 2 \).
The y-intercept is not affected by the slope; it is an independent value that shows a fixed vertical position. Even if the slope changes, the y-intercept remains constant for a given equation unless the equation itself is altered.

• Key takeaway: The y-intercept \( b \) identifies the line's position along the y-axis.
• Practical tip: Always check the y-intercept to visually anchor the line when graphing.

By pinpointing the y-intercept, you gain a clear visual start for any linear equation on a graph.

Slope

The slope of a linear equation, denoted by \( m \), is a measure of how steep a line is. In the slope-intercept form \( y = mx + b \), \( m \) signifies how much \( y \) changes for a unit change in \( x \). If \( m = -2 \), as in this exercise, it indicates that for each increase by 1 unit in \( x \), \( y \) decreases by 2 units.
The slope tells us the direction and angle of the line on the graph. A positive slope means the line ascends from left to right, while a negative slope means the line descends. A zero slope represents a horizontal line, while an undefined slope represents a vertical line.

• Understanding slope is key to determining the line's direction on a graph.
• A steep slope signifies a fast rate of change, whereas a gentle slope indicates a slower rate.
• The slope is calculated by the ratio \( \frac{\text{rise}}{\text{run}} \).

In our equation, \( m = -2 \), the line tilts downwards, reflecting a negative relationship between \( x \) and \( y \). This insight helps interpret and graph linear equations effectively.