Problem 18 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((1,5)\) and \((3,11)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 3x + 2\).

Step 1 - Find the Slope

To write the equation of the line, first find the slope using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \(x_1 = 1, y_1 = 5, x_2 = 3, y_2 = 11\). Substitute these values in to get: \[ m = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3 \]

Step 2 - Use the Point-Slope Form

Now, use the point-slope form of the equation of a line which is \[ y - y_1 = m(x - x_1) \]With \(m = 3, x_1 = 1, y_1 = 5\), substitute the values into the equation to get: \[ y - 5 = 3(x - 1) \]

Step 3 - Simplify the Equation

Finally, simplify the equation to put it in the slope-intercept form \[ y = mx + b \]First, distribute the 3 on the right-hand side: \[ y - 5 = 3x - 3 \]Then, add 5 to both sides: \[ y = 3x + 2 \]

Key Concepts

slope calculation

The slope of a line measures how steep it is. In more technical terms, the slope (m) is the measure of the amount of change in the y-value per unit change in the x-value between two points on a line.
To calculate the slope between two points, we use the formula: \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \)
For our problem with points \( (1,5) \) and \( (3,11) \):

• \( x_1 = 1 \), \( y_1 = 5 \)
• \( x_2 = 3 \), \( y_2 = 11 \)

Substitute these values into the formula to find the slope:
\( m = \frac{{11 - 5}}{{3 - 1}} = \frac{{6}}{{2}} = 3 \)
So, the slope of our line is 3. This tells us that for every 1 unit increase in x, the y-value increases by 3 units.

point-slope form

The point-slope form of a line is very useful when you know the slope and a point on the line. This form is written as: \( y - y_1 = m(x - x_1) \)
Here, m is the slope, and \( (x_1, y_1) \) is a given point on the line.
Based on our problem, we have:

• \( m = 3 \)
• \( x_1 = 1 \), \( y_1 = 5 \)

We substitute these values into the point-slope form to get: \( y - 5 = 3(x - 1) \)
This represents the equation of our line, but this form can be simplified further for other uses.

slope-intercept form

The slope-intercept form is another way to write the equation of a line. This form is particularly easy to read because it directly shows the slope and the y-intercept. It is written as: \( y = mx + b \)
Here, m is the slope and b is the y-intercept (the point where the line crosses the y-axis).

To convert our point-slope form equation \( y - 5 = 3(x - 1) \) into slope-intercept form, we follow these steps:

• Distribute the slope on the right-hand side: \( y - 5 = 3x - 3 \)
• Add 5 to both sides of the equation to isolate y: \( y = 3x - 3 + 5 \)
• Simplify the equation: \( y = 3x + 2 \)

Now, the equation is in slope-intercept form \( y = 3x + 2 \). Here, m (the slope) is 3 and b (the y-intercept) is 2.
By knowing how to work with these forms, you can easily find the equation of any line, given a few basic pieces of information.