Problem 32 - Elementary Algebra

Question

Determine the slope and \(y\) -intercept of the lines. $$ y=-x+2 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Equation's Format

: The given equation is already in the form of \(y = mx + b\): $$ y = -x + 2 $$

Step 2: Determine the Slope (m)

: In the equation, the coefficient of the \(x\) term represents the slope: -1 So, the slope (m) is: $$ m = -1 $$

Step 3: Determine the y-intercept (b)

: In the equation, the constant term represents the y-intercept. In this case, the constant term is 2. So, the y-intercept (b) is: $$ b = 2 $$

Step 4: Present the Solution

: The slope of the given line is \(m = -1\), and the y-intercept of the given line is \(b = 2\).

Key Concepts

Slope

The slope in a linear equation is crucial because it tells us how steep a line is. Imagine it as the rate at which something changes. For instance, if you’re walking up a hill, the slope would measure how steep that hill is. In the linear equation format, specifically in the form \(y = mx + b\), the letter \(m\) represents the slope.

This slope is the number that tells you how much \(y\) changes for a unit change in \(x\). In our example equation \(y = -x + 2\), the coefficient of \(x\) is \(-1\).

• The slope \(m = -1\) means for every increase by 1 in \(x\), \(y\) decreases by 1, suggesting a downward trend.
• If the slope were positive, it would suggest an ascent.

Understanding slopes is much like understanding speeds on a highway—negative slopes go downhill, while positive ones go uphill!

y-intercept

The \(y\)-intercept gives us the starting point of the line on the graph, where it crosses the \(y\)-axis. In simpler terms, it is where the line would intersect the vertical axis if the entire graph were to be drawn. The \(y\)-intercept helps you see the point where \(x = 0\).

In the equation format \(y = mx + b\), \(b\) stands for the \(y\)-intercept. For our given equation \(y = -x + 2\),

• The \(y\)-intercept \(b = 2\), indicating the line crosses the \(y\)-axis at the point \((0, 2)\).

Knowing the \(y\)-intercept provides a quick snapshot of the line's positioning on a graph and helps start sketching it without intricate calculations.

Equation Format

Linear equations often come in the form \(y = mx + b\). This is the standard slope-intercept form that makes it easier to find both the slope (\(m\)) and the \(y\)-intercept (\(b\)) directly by looking at the equation.

The equation format provides a clear blueprint where:

• \(m\) tells us the slope.
• \(b\) shows us the \(y\)-intercept.

For the exercise with the equation \(y = -x + 2\), it is already in this friendly and familiar form. Understanding and using this equation format is like assembling a puzzle; each component fits neatly into place to reveal the line's behavior and position in a coordinate system.

When working with linear equations, this format offers the simplest way to quickly identify a line's characteristics in a graphical representation.