Problem 15 - Calculus Single Variable

Question

Write the slope-intercept equation of the line determined by the given data. Slope \(\sqrt{2}, y\) -intercept \(-\sqrt{3}\)

Step-by-Step Solution

Answer

The equation is \( y = \sqrt{2}x - \sqrt{3} \).

Step 1: Identify the Slope-Intercept Form

The slope-intercept form of a line's equation is given by: \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept.

Step 2: Substitute the Values

Given that the slope \( m \) is \( \sqrt{2} \) and the y-intercept \( b \) is \(-\sqrt{3} \), substitute these values into the slope-intercept formula.

Step 3: Write the Final Equation

Substituting \( m = \sqrt{2} \) and \( b = -\sqrt{3} \) into the equation, we have: \( y = \sqrt{2}x - \sqrt{3} \).

Key Concepts

Equation of a Line

The equation of a line plays a fundamental role in algebra and coordinate geometry. A line in a two-dimensional space can be defined by its slope and its intercepts with the axes.
In the slope-intercept form, this equation is expressed as \( y = mx + b \), where 'm' denotes the slope and 'b' symbolizes the y-intercept. This form is particularly user-friendly because it instantly reveals both the rate of change of the line and where it crosses the y-axis.

• **Slope (m)**: Indicates the steepness or incline of the line.
• **Y-intercept (b)**: Identifies the point where the line intersects the y-axis.

By substituting the specific values for 'm' and 'b' into this formula, one can effortlessly write down the equation for any particular line.

Slope

The slope of a line in mathematical terms is essentially its steepness. It expresses how much 'y' changes for a change in 'x'.
Mathematically, it is captured in the formula \( m = \frac{\Delta y}{\Delta x} \), indicating the change in y-value over a change in the x-value. Imagine you are hiking up a hill. The steeper the hill, the higher the slope.
Here are some quick facts about slopes:

• A **positive slope** means the line rises as it moves from left to right.
• A **negative slope** indicates the line falls as it moves from left to right.
• A slope of **zero** implies a horizontal line, showing no rise or fall.
• An **undefined slope** means the line is vertical.

In the example above, the slope is \( \sqrt{2} \), which signifies a moderate incline upwards.

Y-intercept

The y-intercept is another critical aspect of the line's equation. It specifies where exactly the line crosses the y-axis.
In the slope-intercept equation \( y = mx + b \), the 'b' represents this intercept. If you were to start drawing the line from this point on the y-axis, it's the very first location you'd plot.

• It gives the starting point of the line when \( x = 0 \).
• The y-intercept can be a positive or negative number, influencing whether the line begins above or below the x-axis.

In the given equation \( y = \sqrt{2}x - \sqrt{3} \), the y-intercept is \( -\sqrt{3} \), pinpointing an initial downward crossing on the y-axis.