Problem 4 - Algebra A Combined Function

Question

Find the slope of the line that passes through the given points. See Examples 1 and 2. $$ (3,1) \text { and }(2,6) $$

Step-by-Step Solution

Answer

The slope of the line is -5.

Step 1: Understanding the Slope Formula

The slope of a line passing through two points, say \( (x_1, y_1) \) and \( (x_2, y_2) \), is calculated using the following formula: \[slope, \, m = \frac{y_2 - y_1}{x_2 - x_1}.\]For our points (3,1) and (2,6), identify \( x_1 = 3, \, y_1 = 1, \, x_2 = 2, \, ext{and} \, y_2 = 6. \)

Step 2: Subtract the y-coordinates

Using the formula, first subtract the y-coordinates:\[y_2 - y_1 = 6 - 1 = 5.\]

Step 3: Subtract the x-coordinates

Next, subtract the x-coordinates:\[x_2 - x_1 = 2 - 3 = -1.\]

Step 4: Calculate the Slope

Now, substitute the differences back into the slope formula:\[m = \frac{5}{-1} = -5.\]So, the slope of the line is -5.

Key Concepts

Coordinate Geometry

Coordinate geometry is a fascinating branch of geometry where we use a coordinate system to study the spatial relationships and properties of points, lines, and figures. Imagine a flat surface, like a piece of graph paper, marked with horizontal and vertical lines which intersect at a point called the origin. This setup helps us pinpoint any spot on the plane using coordinates, usually in the form of

• an x-coordinate (horizontal position)

and

• a y-coordinate (vertical position).

When you know the coordinates of two points in this system, you can find many things about the line they form. One prime property of such a line is the 'slope.' In real life, this is like understanding how steep a hill is when you have two spots on it. So, in the exercise provided, coordinate geometry involves identifying these two points:

• (3,1)

and

• (2,6)

,
then using the slope formula to determine how steep the line connecting them is.

Slope Formula

The slope formula is a handy tool in mathematics and is crucial for understanding how lines behave. The slope of a line is essentially the measure of how steep the line is, and it tells us how much the y-coordinate (vertical) of a point on a line changes for a unit change in the x-coordinate (horizontal).The formula to calculate slope, denoted usually by 'm,' when we have two known points on the line, say

• \((x_1, y_1)\)
• and \((x_2, y_2)\)

is:\[m = \frac{y_2 - y_1}{x_2 - x_1} \]In our example, you use it to find that the slope of the line passing through the points

• (3,1)
• and (2,6)

is -5. This means that with every single unit you move right along the line (an increase on the x-axis), you move down five units (a decrease on the y-axis).Understanding this calculation can help in fields like physics, where we may be interested in the rate of change of certain quantities, or even in finance, where we might be trying to understand the rate of growth or decline of a value over time.

Linear Equations

Linear equations are mathematical expressions equating two algebraic expressions that result in a straight line when graphed on the Cartesian plane. The standard form of a linear equation in two variables x and y is given as \(y = mx + c\), where:

• m represents the slope,
• c is the y-intercept,

which indicates where the line intersects the y-axis.For the exercise provided, you can think of the linear equation as a way to describe the line connecting the points

• (3,1)
and
• (2,6)

in the form \(y = -5x + c.\)With the calculated slope of -5, you can plot this line if you also determine the y-intercept. This equation allows you to predict the y-coordinate for any x-coordinate, thus forming a complete relationship achieved through coordinate geometry fundamentals.Linear equations serve as a building block in algebra and calculus, helping us model and solve real-world problems, from predicting costs over time to understanding trends in data.