Problem 2 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((1,4)\) and \((7,6)\)

Step-by-Step Solution

Answer

The slope is 0.33.

Step 1: Identify the given points

The given points are \(x_{1}, y_{1}\) = (1, 4) and \(x_{2}, y_{2}\) = (7, 6).

Step 2: Recall the formula for the slope

The slope of a line passing through two points \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \) is given by the formula: \[ m = \frac{{y_{2} - y_{1}}}{{x_{2} - x_{1}}} \]

Step 3: Substitute the coordinates into the formula

Using the points (1, 4) and (7, 6), substitute into the formula to get: \[ m = \frac{{6 - 4}}{{7 - 1}} \]

Step 4: Simplify the expression

Calculate the numerator and denominator: \[ m = \frac{2}{6} = \frac{1}{3} \]

Step 5: Round to the nearest hundredth if necessary

Since \( \frac{1}{3} \) is approximately 0.33 when rounded to the nearest hundredth.

Key Concepts

coordinate geometry

Coordinate geometry is a branch of mathematics that uses algebraic techniques to study geometric problems. It involves plotting points on a plane using coordinates, which are pairs of numbers like \((x, y)\). These coordinates define the position of points in a Cartesian plane. When we know the coordinates of two points, we can find distances between them, midpoints, and slopes, among other things.
For example, given the points \((1, 4)\) and \((7, 6)\), we can plot these points on a plane. The x-coordinates are 1 and 7, and the y-coordinates are 4 and 6. These pairs represent specific locations on the plane. Understanding coordinate geometry is essential for solving problems related to lines and shapes in a 2D space.

linear equations

Linear equations are algebraic expressions that represent straight lines in a coordinate plane. They can be written in different forms, such as the slope-intercept form, which is \ y = mx + b \, where \ m \ is the slope and \ b \ is the y-intercept.
To find the slope of a line passing through two points, such as the points \((1, 4)\) and \((7, 6)\), we use the formula for slope: \ m = \frac{{y_{2} - y_{1}}}{{x_{2} - x_{1}}} \. This formula gives us the rate at which the line rises or falls as we move from one point to another.
By substituting the values of the points into the formula, \ m = \frac{{6 - 4}}{{7 - 1}} \, we find that the slope \ m = \frac{2}{6} = \frac{1}{3} \.

algebraic expressions

Algebraic expressions are mathematical statements that include numbers, variables, and operations. Expressions can represent quantities in a general form, making them essential tools for solving equations.
In the context of finding the slope of a line, the expression \ m = \frac{{y_{2} - y_{1}}}{{x_{2} - x_{1}}} \ is an algebraic way to describe the slope. Here, \ y_{2} - y_{1} \ and \ x_{2} - x_{1} \ are differences in the y and x coordinates, respectively.
Simplifying this expression by performing the operations will give us the slope. For our points \((1, 4)\) and \((7, 6)\), substituting in and simplifying \ m = \frac{2}{6} = \frac{1}{3} \ shows us that the slope is approximately 0.33. Understanding algebraic expressions is key to manipulating and solving different mathematical problems.