Problem 51 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through (2,3) and (0,0)

Step-by-Step Solution

Answer

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

Step 1: Understand the Slope Formula

To find the equation of a line, we first need to determine its slope. The slope (m) of a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\), is calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 2: Calculate the Slope

Identify the points given: \((2,3)\) and \((0,0)\). Applying these points to the slope formula, we get: \( m = \frac{3 - 0}{2 - 0} = \frac{3}{2} \). The slope of the line is \( \frac{3}{2} \).

Step 3: Write the Equation in Point-Slope Form

Using the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \), select one point, for example, \((0,0)\). Plug in the values: \( y - 0 = \frac{3}{2}(x - 0) \), which simplifies to \( y = \frac{3}{2}x \).

Step 4: Convert to Slope-Intercept Form

The equation \( y = \frac{3}{2}x \) is already in slope-intercept form because it matches \( y = mx + b \), where \( m = \frac{3}{2} \) and \( b = 0 \).

Key Concepts

Slope of a Line

The slope of a line is a measure that describes its steepness and direction. Imagine you're hiking up a hill: the steeper the hill, the higher the slope. Mathematically, the slope is defined as the ratio of the change in the vertical direction (\(\Delta y\)) to the change in the horizontal direction (\(\Delta x\)). This is represented by the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line.
If the slope is positive, the line ascends from left to right. Conversely, if the slope is negative, it descends. A slope of zero indicates a horizontal line, while an undefined slope represents a vertical line.
In our exercise, we used two points, \((2,3)\) and \((0,0)\). By substituting these into the slope formula, we obtained \(m = \frac{3}{2}\), indicating a line that rises as it moves to the right.

Point-Slope Form

The point-slope form is a powerful way to express the equation of a line, especially useful when you know the slope of the line and a point on the line. It is written as:

• \(y - y_1 = m(x - x_1)\)

Here, \(m\) represents the slope, and \((x_1, y_1)\) is a specific point on the line.
To write the equation using the point-slope form, simply plug in the known values for \(m\), \(x_1\), and \(y_1\).
In our example, selecting the point \((0,0)\) and using the calculated slope \(m = \frac{3}{2}\), we substitute these values into the point-slope equation:

• \(y - 0 = \frac{3}{2}(x - 0)\)

This simplifies straightforwardly to \(y = \frac{3}{2}x\). The simplicity of this form aids in quickly finding the equation, as demonstrated.

Equation of a Line

Writing the equation of a line can be done in several forms, but perhaps the most common is the slope-intercept form, which is particularly easy to understand and use. The slope-intercept form is expressed as:

• \(y = mx + b\)

Here, \(m\) is the slope of the line, and \(b\) represents the y-intercept, the point where the line crosses the y-axis.
It's efficient because it shows both the slope and the y-intercept directly in one equation.
In our exercise, after using the point-slope form, we easily identified that our line equation \(y = \frac{3}{2}x\) is already in slope-intercept form, with the slope \(m = \frac{3}{2}\) and the y-intercept \(b = 0\).
This form provides clear and immediate information about the line's characteristics, making graphing straightforward and solving other related problems easier.