Problem 31 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ y=-x+2 $$

Step-by-Step Solution

Answer

Answer: The slope of the linear equation is -1, and the y-intercept is 2.

Step 1: 1. Identify the slope and y-intercept

First, take a look at the given equation: $$ y = -x + 2 $$ Here, you can see that it's already in the slope-intercept form. From this, we can directly identify the slope and \(y\)-intercept.

Step 2: 2. Slope

In the slope-intercept form, the slope (\(m\)) is the coefficient of the \(x\) term. So, in our equation: $$ y = -x + 2 $$ the slope (\(m\)) is -1.

Step 3: 3. y-intercept

In the slope-intercept form, the \(y\)-intercept (\(b\)) is the constant term. So, in our equation: $$ y = -x + 2 $$ the \(y\)-intercept (\(b\)) is 2.

Step 4: 4. Conclusion

For the given linear equation: $$ y = -x + 2 $$ the slope (m) is -1, and the y-intercept (b) is 2.

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that describe a straight line on a graph. They are often written in the form \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. This format is known as the slope-intercept form. Linear equations are called so because their graph results in a line, hence the name "linear."Key features of linear equations include:

• They have a constant rate of change, which means the slope does not vary across the line.
• They can model direct relationships, such as distance over time or cost over quantity.
• The simplicity of these equations allows for ease in graphing and solving real-world problems.

Being able to manipulate these equations into the slope-intercept form is crucial. This form makes it straightforward to analyze and graph the line, identifying key features like the slope and the point where the line crosses the y-axis.

Decoding the Slope

The slope of a line in a linear equation is a measure of its steepness or tilt. It is symbolized by \(m\) in the equation \(y = mx + b\). For the equation you're given, \(y = -x + 2\), the slope is \(-1\), since there is an implied coefficient of \(-1\) in front of the \(x\).To better understand slope, think of it as "rise over run":

• "Rise" refers to the vertical change (up or down movement).
• "Run" refers to the horizontal change (left or right movement).

So, a slope of \(-1\) means that for every one unit you move to the right on the \(x\)-axis, you move one unit down on the \(y\)-axis. A positive slope would mean the line rises as it moves right, whereas a negative slope means it falls.Understanding slope is important not only for graphing lines, but also for interpreting the direction and speed at which a variable changes relative to another.

The Beauty of the Y-Intercept

The \(y\)-intercept of a linear equation is where the line crosses the y-axis on a graph. In the equation \(y = mx + b\), the term \(b\) represents the y-intercept. For the equation \(y = -x + 2\), the y-intercept is \(2\).This means:

• When \(x\) is zero, \(y\) will be \(2\).
• Graphically, it's the point (0, 2) on the coordinate plane.

Understanding the y-intercept helps in quickly plotting the starting point of the line on a graph. It's also crucial for interpreting initial values in real-world scenarios. For example, if \(y\) represents money and \(x\) time, the y-intercept tells us the starting amount of money before any activity occurs.Thus, knowing the y-intercept not only aids in graphing but also provides meaningful insights into the context of problems modeled by linear equations.