Problem 23 - 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. $$ (-1,-2),(3,-2) $$

Step-by-Step Solution

Answer

The line that passes through the points (-1,-2) and (3,-2) has an equation of \(y = -2\).

Step 1: Plotting the points

Plot the two points (-1,-2) and (3,-2) on the Cartesian plane. We observe that both points lie on the same horizontal line at y = -2.

Step 2: Drawing the line through points

Next, draw a straight line that passes through the two points. The line will be horizontal as both the points have the same y-coordinate.

Step 3: Finding the slope

The slope, m, of a line that passes through points (x1, y1) and (x2, y2) is given by the formula \(m = \frac{y2 - y1}{x2 - x1}\). Substituting our given points into this formula will give \(m = \frac{-2 - (-2)}{3 - (-1)} = 0\).

Step 4: Writing the equation

The equation of a line in slope-intercept form is given by y = mx + b, where m is the slope and b is the y-intercept. The y-intercept for our line is the y coordinate of any point on the line, which is -2. Hence, the equation is \(y = 0x -2\) or \(y = -2\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express the equation of a line. It is written as \(y = mx + b\). This formula provides a simple way to see at a glance the two key characteristics of a line: its slope \(m\) and its y-intercept \(b\). The slope \(m\) indicates how much the line rises or falls as it moves from left to right. A positive slope rises, while a negative slope falls. The y-intercept \(b\) is the point where the line crosses the y-axis, indicating where the line sits vertically. In the given equation \(y = mx + b\), \(m\) forms the coefficient that determines the angle of tilt, and \(b\) tells you at which point the line hits the y-axis. Understanding this form is crucial to quickly sketch a line or understand its key properties without needing to graph it every time.

Graphing Points

Graphing points is a fundamental skill for plotting linear equations on a Cartesian plane. The Cartesian plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). When graphing a point, it is represented as \((x, y)\). The x-coordinate tells you how to move horizontally from the origin, while the y-coordinate indicates how to move vertically. For example, to plot \((-1, -2)\), move 1 unit left along the x-axis, then 2 units down along the y-axis. Plot another point, such as \((3, -2)\), by moving 3 units right and then 2 units down. These points lie on a flat line when the y-coordinates are identical, showing a horizontal line. By understanding how to plot points, you can visualize equations and better understand their graphical properties.

Equation of a Line

The equation of a line is a mathematical expression that describes all the points that make up the line. With just a little information—typically a slope and a y-intercept—you can write an equation that represents a line in slope-intercept form, \(y = mx + b\). If the slope \(m\) is 0, as in our example above, the line is horizontal because there is no vertical change as you move along it. This is why the equation simplifies to \(y = b\). In the example with the points \((-1, -2)\) and \((3, -2)\), both points share the same y-coordinate \(-2\), making the slope 0 and the line horizontal. Thus, the equation of the line becomes \(y = -2\). Recognizing such line behaviors helps you understand not only how a line looks but how it functions within a coordinate system.