Problem 34 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the line using the given information. The slope is \(\frac{3}{4}\) and it passes through the point (1,4)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{3}{4}x + \frac{13}{4} \).

Step 1: Understand the slope-intercept form

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

Step 2: Identify given values

We are given the slope \( m = \frac{3}{4} \) and a point \( (x_1, y_1) = (1, 4) \) that lies on the line.

Step 3: Substitute slope into the equation

Substitute the given slope into the equation \( y = mx + b \) to get \( y = \frac{3}{4}x + b \).

Step 4: Use the point to solve for b

Substitute the coordinates of the point into the equation: \( 4 = \frac{3}{4}(1) + b \). This simplifies to \( 4 = \frac{3}{4} + b \).

Step 5: Solve for y-intercept b

To find \( b \), subtract \( \frac{3}{4} \) from both sides: \( b = 4 - \frac{3}{4} \). This simplifies to \( b = \frac{16}{4} - \frac{3}{4} = \frac{13}{4} \).

Step 6: Write the final equation

Substitute \( b = \frac{13}{4} \) back into the slope-intercept form: \( y = \frac{3}{4}x + \frac{13}{4} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common forms used to describe a straight line in the coordinate plane. This form makes it easy to see and understand both the slope of the line and where it crosses the y-axis. The general formula for the slope-intercept form is \( y = mx + b \). In this equation format:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept of the line, or the point where the line crosses the y-axis.

The purpose of this form is to allow quick identification of these two parameters. For instance, if you're given \( y = 2x + 3 \), you can immediately see that the slope \( m \) is 2, and the line crosses the y-axis at the point \( (0, 3) \). Understanding this form is crucial, as it lays the foundation for creating and interpreting equations that describe lines.

Point-Slope Form

The point-slope form offers a different starting point for writing the equation of a line, useful when you know a line's slope and a particular point on the line. This form is particularly helpful when the y-intercept is not readily available. The formula for point-slope form is given by: \[ y - y_1 = m(x - x_1) \] Here:

• \( (x_1, y_1) \) is a specific point on the line known as a point of passage.
• \( m \) is the slope of the line, representing how steep the line is.

To illustrate, imagine you are given the point (3, 2) and a slope of 4. You plug these values into the formula to get \( y - 2 = 4(x - 3) \). From here, you can solve for \( y \) to see the line in slope-intercept form. This transformation allows you to perceive both the location and inclination of the line swiftly.

Finding the Equation of a Line

Finding the equation of a line generally involves using either the slope-intercept form or the point-slope form. When you have a specific point along with the slope of a line, you can effectively put these pieces together to form the equation.Here’s how you can use the given information to find a line's equation:**Step 1: Identify the Given Information** Start by recognizing the slope and a point through which the line passes. Suppose you have \( m = \frac{3}{4} \) and the point \( (1, 4) \).**Step 2: Use the Slope-Intercept Form** Apply the slope-intercept format by substituting \( m \) into \( y = mx + b \), which gives \( y = \frac{3}{4}x + b \).**Step 3: Solve for the Y-Intercept \( b \)** Use the provided point, \( (1, 4) \), and plug it into the equation. Solve \( 4 = \frac{3}{4}(1) + b \) to find \( b \), which results in \( b = \frac{13}{4} \).**Step 4: Write the Final Equation** Substitute back \( b = \frac{13}{4} \) into the equation to get \( y = \frac{3}{4}x + \frac{13}{4} \).This method of finding the line's equation is integral in algebra and geometry, as it seamlessly connects the slope and specific points to form a comprehensive line equation.