Problem 34 - College Algebra

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 is \( y = \frac{3}{4}x + \frac{13}{4} \).

Step 1: Identify the Given Information

We are given the slope of the line, which is \( \frac{3}{4} \), and a point that the line passes through, \((1,4)\). Our task is to find the equation of the line using this information.

Step 2: Use Point-Slope Formula

The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line. Substituting \( m = \frac{3}{4} \), \( x_1 = 1 \), and \( y_1 = 4 \), we get: \[ y - 4 = \frac{3}{4}(x - 1) \]

Step 3: Simplify the Equation

Expand the right side of the equation: \[ y - 4 = \frac{3}{4}x - \frac{3}{4} \]Now, solve for \( y \):\[ y = \frac{3}{4}x - \frac{3}{4} + 4 \]Combine like terms:\[ y = \frac{3}{4}x + \frac{16}{4} - \frac{3}{4} \]This simplifies to:\[ y = \frac{3}{4}x + \frac{13}{4} \]

Step 4: Write the Final Equation

The final equation of the line in slope-intercept form is:\[ y = \frac{3}{4}x + \frac{13}{4} \]

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line, and it's particularly user-friendly. It's written as \( y = mx + b \), where:

• \( m \) represents the slope of the line. This tells us how steep the line is, or how much \( y \) changes with a change in \( x \).
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

By using the slope-intercept form, you can easily find out key details about the line just by looking at the equation. For example, in the equation \( y = \frac{3}{4}x + \frac{13}{4} \), the slope is \( \frac{3}{4} \), indicating that for every 4 units increase in \( x \), \( y \) will increase by 3 units. The y-intercept is \( \frac{13}{4} \), meaning the line crosses the y-axis at this point. This form is particularly beneficial when graphing linear equations, as you can swiftly sketch the graph knowing the slope and y-intercept without intricate calculations.

Point-Slope Formula

The point-slope formula is particularly useful when you know one point on a line and the slope of that line. The formula is given by:\[ y - y_1 = m(x - x_1) \] Here:

• \( m \) is the slope of the line.
• \((x_1, y_1)\) is a known point on the line.

To find the equation of a straight line with a known slope and point, you substitute the given values into this formula. In the example, with a slope of \( \frac{3}{4} \) and a point \((1, 4)\), you'd start with:\[ y - 4 = \frac{3}{4}(x - 1) \]By distributing \( \frac{3}{4} \) and then simplifying, you can rearrange to obtain the equation in slope-intercept form. Using the point-slope formula can also help verify solutions by checking if a particular point lies on the line described by the equation.

Linear Equations

Linear equations represent straight lines in a two-dimensional graph. These equations are typically expressed in forms like slope-intercept or point-slope, each providing distinct advantages for different situations.
Linear equations can be identified due to the degree of variables being one; this forms the graph of a straight line. They are simple yet powerful for modeling relationships with consistent rates of change.

Features of Linear Equations

• Constant slope: The rate of change between any two points is the same.
• Degree of one: Each variable is raised to the power of one, reflecting a direct proportionality.
• Graphically represented as a straight line: This simplicity allows for straightforward graphical interpretation.

In the context of a real-world scenario, linear equations can model relationships like distance over time at a constant speed, or cost per item in a purchase. Understanding linear equations form the foundation for exploring more complex functions, making them an essential concept in algebra.