Problem 39 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ (0,0),(1,1) $$

Step-by-Step Solution

Answer

The equation of the line is: y = x.

Step 1: Determine the slope of the line

To find the slope (m), we can use the formula: $$ m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} $$ We already have the two points: \((0,0)\) and \((1,1)\). Assign the values to their respective coordinates: \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) : $$ x_{1} = 0, y_{1} = 0 \\ x_{2} = 1, y_{2} = 1 $$ Substitute these values into the slope formula: $$ m = \frac{1 - 0}{1 - 0} = \frac{1}{1} = 1 $$

Step 2: Determine the y-intercept

The y-intercept (b) is the point where the line intersects the y-axis. In this case, the line passes through the origin \((0,0)\), so the y-intercept is 0.

Step 3: Write the equation in slope-intercept form

Now that we have the slope (m = 1) and the y-intercept (b = 0), we can write the equation of the line in slope-intercept form (\(y = mx + b\)): $$ y = 1x + 0 $$ To simplify, we can omit "+0": $$ y = x $$ So, the equation of the line passing through \((0,0)\) and \((1,1)\) is: $$ y = x $$

Key Concepts

Equation of a Line

An equation of a line describes all the points that form a particular line on a graph. In the slope-intercept form, this equation is expressed as \( y = mx + b \).
Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis.
The simplicity of the slope-intercept form makes it a popular choice for graphing linear equations and quickly finding both the slope and y-intercept at a glance.

• **Slope (\( m \))**: Indicates the steepness and direction of the line.
• **Y-intercept (\( b \))**: Tells us where the line crosses the y-axis.

Understanding the equation of a line helps in predicting how changes in one variable affect another.
This equation is fundamental in both algebra and coordinate geometry, explaining how different lines behave depending on their slope and intercept.

Slope

The slope of a line quantifies its steepness and direction. In our example, the slope is found using two specific points on the line, \((0,0)\) and \((1,1)\).
To calculate the slope \( m \), we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]The difference in the y-values is divided by the difference in the x-values.
This formula helps us measure how much the line rises or falls as we move along the x-axis.

Example Calculation

Given the points \((0,0)\) and \((1,1)\):

• \( x_1 = 0, y_1 = 0 \)
• \( x_2 = 1, y_2 = 1 \)

Substituting these into the formula gives:\[ m = \frac{1 - 0}{1 - 0} = 1 \]This means the line rises one unit for every one unit it runs to the right, indicating a 45-degree angle with the x-axis.
This positive slope means the line moves upwards as it extends to the right.

Y-intercept

The y-intercept is the point where the line crosses the y-axis. It is represented by \( b \) in the slope-intercept form of the equation \( y = mx + b \). A line's y-intercept gives a starting point from which we can set the direction, as determined by the slope.

In our problem, the y-intercept is 0 because the line passes through the origin, \((0,0)\).

• A y-intercept of 0 means that at \( x = 0 \), \( y \) is also 0. The line directly starts from the origin.
• The y-intercept is crucial because it provides a reference point to draw the line accurately given its slope.

Thinking of the y-intercept as the "starting point" on the y-axis helps to visualize where a line begins before it stretches out and up, or down, across the graph influenced by the slope.
In various contexts, it's also useful for interpreting real-world situations, like determining an initial quantity or condition when analyzing data trends.