Problem 85 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ (0,2), \quad(1,0) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = -2x + 2.

Step 1: Find the slope

To find the slope of the line, we will use the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ where (x1, y1) and (x2, y2) are the given points. Plugging in given points (0,2) and (1,0), we get: $$ m = \frac{0 - 2}{1 - 0} = \frac{-2}{1} = -2 $$

Step 2: Find the y-intercept

Since one of the given points is the y-intercept itself (0,2), the y-intercept (b) is 2.

Step 3: Write the equation in slope-intercept form

We have the slope (m = -2) and the y-intercept (b = 2). Now, we will write the equation in slope-intercept form: $$ y = mx + b $$ Plugging in the values of m and b, we get: $$ y = -2x + 2 $$

Key Concepts

Linear Equations

Linear equations describe a straight line when graphed on a coordinate plane. They are fundamental in algebra and are represented in various forms, including the slope-intercept form.
The general form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. Although linear equations can take many forms, we often convert them to simplify our calculations.
In problems requiring the equation of a line from two points, converting to the slope-intercept form makes it easier to identify key characteristics of the line, such as the slope and y-intercept. This form is particularly popular in algebra because it directly shows how the line behaves as it crosses the y-axis and how steep the line is (its slope).

Finding Slope

The slope of a line is a measure of its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between two distinct points on the line.
To find the slope, we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line. The letter \(m\) is commonly used to represent the slope.
For example, using the points \((0, 2)\) and \((1, 0)\), the slope \(m\) can be calculated as \(-2\). Since a negative slope indicates the line is falling as it moves from left to right, you can visualize the impact of such a slope in a downward trend on the graph.
Finding the slope is a crucial step in crafting the equation of a line, particularly when writing in slope-intercept form.

y-Intercept

The y-intercept is the point where the line crosses the y-axis. This is the value of \(y\) when \(x\) is equal to zero.The y-intercept is vital because it provides a starting point on the graph for the line. Understanding this concept helps in plotting lines and interpreting the context of linear relationships in real-life scenarios.
For a linear equation in slope-intercept form \(y = mx + b\), \(b\) is the y-intercept.

• In our example, one of the given points \((0, 2)\) is the y-intercept, where \(b = 2\).

This point offers a direct way to graph the equation and aids in understanding where the line will cut through the y-axis.
By knowing both slope and y-intercept, constructing the full equation becomes straightforward, leading to complete understanding and effective manipulation of linear equations in different situations.