Problem 24 - Algebra 1

Question

Graph the points and draw a line through them. Write an equation in slope- intercept form of the line that passes through the points. $$ (2,0),(-2,6) $$

Step-by-Step Solution

Answer

The slope-intercept form of the line passing through the points (2,0) and (-2,6) is \( y = -1.5x \).

Step 1: Plot the Points

To begin with, plot the points (2,0) and (-2,6) onto the graph.

Step 2: Calculate the Slope

Next, calculate the slope (m) using the formula \( m = (y2 - y1) / (x2 - x1) \). Substituting our given points, (-2,6) as (x2, y2) and (2,0) as (x1, y1), well get \( m = (6 - 0) / (-2 - 2) = -6/4 = -1.5 \). So the slope of the line is -1.5.

Step 3: Form the Equation

Finally, derive the equation of the line using the slope-intercept form, y=mx+b, where m is the slope, and b is the y-intercept. Here, the y-intercept (b) is the y-coordinate of either of the point on the line where the line crosses the y-axis. From the points given, we know (2,0) crosses the y-axis. So, b = 0. Substituting m = -1.5 and b = 0 into our equation gives us the line equation as \( y = -1.5x \).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of geometry where we use a coordinate plane to analyze and represent geometric figures. We represent points using ordered pairs (x, y) on this plane. For example, the point (2,0) means we move 2 units right on the x-axis and stay at the origin on the y-axis.
Coordinate geometry helps in visually understanding relationships between algebraic equations and geometric figures.

• The x-axis is the horizontal line, while the y-axis is the vertical line.
• The origin is the point where both axes intersect, denoted as (0,0).
• Every point on the plane has a unique set of coordinates (x, y).

Coordinate geometry is fundamental when dealing with linear equations, as it allows us to graphically represent these equations to better understand their behavior and relationships.

Linear Equations

Linear equations form straight lines on a coordinate graph. A common form of a linear equation is the slope-intercept form, expressed as \( y = mx + b \). Here, \( m \) is the slope, displaying the steepness and direction of the line, while \( b \) is the y-intercept, indicating where the line crosses the y-axis.

• The slope \( m \) is calculated using two points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \), with the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls.
• The y-intercept \( b \) is found by setting \( x = 0 \) and solving for \( y \) in the equation.

Understanding linear equations is crucial in solving problems related to lines, slopes, and intersections.

Plotting Points

Plotting points on a graph is the process of marking specific locations determined by their coordinates. For instance, to plot the points (2,0) and (-2,6), you need to identify these coordinates on a graph with x and y axes and mark them accordingly.
When plotting points:

• Identify the x-coordinate first and move horizontally.
• Then, move vertically based on the y-coordinate.
• Mark the intersection of these movements with a point.

Plotting helps in visualizing data or solutions to equations. When two points are plotted, you can draw a straight line through them to start visualizing the linear equation they form. This method is hugely beneficial in understanding how equations translate into visual graphs.