Problem 30 - Elementary Algebra

Question

You are given one point on a line and the slope of the line. Find the coordinates of three other points on the line. $$(-2,6), m=-\frac{3}{7}$$

Step-by-Step Solution

Answer

Points: \((-1, \frac{39}{7})\), \((0, \frac{36}{7})\), \((1, \frac{33}{7})\).

Step 1: Understand the Problem

We are given a point \((-2, 6)\) on a line and the slope \(m = -\frac{3}{7}\). Our task is to find the coordinates of three other points on this line.

Step 2: Use the Point-Slope Formula

The point-slope form of a line's equation is \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.Substituting \(x_1 = -2\), \(y_1 = 6\), and \(m = -\frac{3}{7}\), we get:\(y - 6 = -\frac{3}{7}(x + 2)\).

Step 3: Find the Equation in Slope-Intercept Form

To express this equation in slope-intercept form \(y = mx + b\),expand the equation:\(y - 6 = -\frac{3}{7}x - \frac{6}{7}\),which simplifies to\(y = -\frac{3}{7}x + \frac{36}{7}\).

Step 4: Choose Different x-values and Find Corresponding y-values

Select values of x to find other points. Let's choose \(x = -1, 0, \text{ and } 1\).For \(x = -1\): \(y = -\frac{3}{7}(-1) + \frac{36}{7} = \frac{3}{7} + \frac{36}{7} = \frac{39}{7}\). So, point \((-1, \frac{39}{7})\).For \(x = 0\): \(y = -\frac{3}{7}(0) + \frac{36}{7} = \frac{36}{7}\). So, point \((0, \frac{36}{7})\).For \(x = 1\): \(y = -\frac{3}{7}(1) + \frac{36}{7} = -\frac{3}{7} + \frac{36}{7} = \frac{33}{7}\). So, point \((1, \frac{33}{7})\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. This form is written as:\[ y = mx + b \]where:- \( y \) is the dependent variable (or the y-value)- \( m \) is the slope of the line, indicating its steepness- \( x \) is the independent variable (or the x-value)- \( b \) is the y-intercept, which is where the line crosses the y-axis.
The slope-intercept form is particularly useful because from the equation, you can immediately determine both the slope \( m \) and the y-intercept \( b \). This helps in graphing the line quickly or understanding its behavior visually. To convert an equation from point-slope form to slope-intercept form, you expand and rearrange it to solve for \( y \). This way, the equation becomes more straightforward, providing a clear picture of how changes in \( x \) affect \( y \). Using this form, you can easily generate a graph of the line by starting at the y-intercept \( b \) and then using the slope \( m \) to find other points on the line.

Coordinates

Coordinates are the numerical values that describe a specific point's position on a plane. They are written as pairs \( (x, y) \), where:

• \( x \)-coordinate represents the horizontal displacement of the point from the origin
• \( y \)-coordinate represents the vertical displacement of the point from the origin

These pairs tell you exactly where a point lies in a two-dimensional space. In the context of lines, knowing at least one point on a line along with the slope can help you identify or place other points on the same line. For instance, if one point is \((-2, 6)\) and you know that the slope of the line is \(-\frac{3}{7}\), you can determine other points by selecting different \( x \)-values and using them in the line's equation to find the corresponding \( y \)-values. This application of coordinates is fundamental in both graphing and understanding the geometric nature of linear equations.

Linear Equations

Linear equations are algebraic expressions that represent straight lines. The general form of a linear equation in two variables is written as:\[ Ax + By = C \]where:- \( A \), \( B \), and \( C \) are constants.Another common form for linear equations is the slope-intercept form \( y = mx + b \). The relationship between \( x \) and \( y \) is linear, meaning their graph is a straight line. The slope \( m \) determines the angle or tilt of this line, while the y-intercept \( b \) determines where this line crosses the y-axis.
Linear equations are simple yet powerful tools in mathematics because they describe the most straightforward relationships between two variables. By analyzing a linear equation, such as by converting it into the slope-intercept form, you can solve for one variable in terms of the other, plot the line on a coordinate grid, and understand how the change in one variable affects the other. This makes linear equations essential in fields like physics, engineering, economics, and more, where relationships are often directly proportional, as in the case of demand and supply, force and mass, or distance and time.