Problem 31 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. slope \(-0.5,\) passes through \((2,-3)\)

Step-by-Step Solution

Answer

The equation is \( y = -0.5x - 2 \).

Step 1: Understand the Slope-Intercept Form

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

Step 2: Identify Given Values

From the problem, we know that the slope \( m \) is \(-0.5\) and the line passes through the point \( (2, -3) \). We will use these values to find the y-intercept \( b \).

Step 3: Substitute into Slope-Intercept Equation

Substitute the given slope \( m = -0.5 \) and the coordinates of the given point \( (x, y) = (2, -3) \) into the equation \( y = mx + b \). This gives us \( -3 = -0.5 \times 2 + b \).

Step 4: Solve for \( b \)

Calculate the right hand side: \( -0.5 \times 2 = -1 \). So the equation becomes \( -3 = -1 + b \). Solve for \( b \): \( b = -3 + 1 = -2 \).

Step 5: Write Final Equation

With \( m = -0.5 \) and \( b = -2 \), substitute these back into the slope-intercept form equation to get \( y = -0.5x - 2 \).

Key Concepts

Equation of a Line

In mathematics, an equation of a line represents all the points through which a straight line passes. In simple terms, it gives us a way to describe a line on a coordinate plane.
This equation can be written in various forms, but the slope-intercept form is one of the most popular. It neatly organizes the information needed to describe the line's slope and where it crosses the y-axis.
The equation tells us important information at a glance, such as how steep the line is and where it begins on the vertical axis. This clarity makes the equation of a line a valuable tool in both algebra and geometry.

Slope

The slope of a line is a measure of its steepness and direction. It indicates how much a line rises or falls as we move from left to right on a graph.
The slope is represented by the letter "m" in the slope-intercept equation, which is written as \( y = mx + b \). To calculate the slope, you take the change in y-values divided by the change in x-values between two points on the line.

• If the slope is positive, the line rises as it moves from left to right.
• If negative, like in our original problem, the line falls as it moves from left to right.
• A slope of zero indicates a horizontal line, and an undefined slope corresponds to a vertical line.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis of a graph. It's crucial because it shows the value of y when x equals zero.
We use "b" to represent the y-intercept in the slope-intercept equation \( y = mx + b \). It's the starting point of the line on the graph.
To find the y-intercept, especially when given a point and a slope, we insert these values into the equation and solve for "b". For instance, in our exercise, this involved solving \( -3 = -0.5 \times 2 + b \), ultimately finding that \( b = -2 \). This y-intercept shows exactly where our line hits the y-axis.

Linear Equations

Linear equations describe relationships that involve straight lines. These relationships are essential in understanding how quantities vary with one another.
Linear equations in two variables can be depicted using graphs and appear as straight lines.
Some characteristics of linear equations:

• They have constant slopes, meaning the rate of change is uniform throughout.
• They are formulated with no terms involving powers or exponents higher than one.
• They typically appear in the familiar form of \( y = mx + b \), where both the slope and y-intercept can be easily identified.

Understanding linear equations helps in solving real-life problems, like determining the trajectory of a moving object or the steady growth of investments over time.