Problem 1 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=3,\) passing through \((2,5)\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y=3x-1\).

Step 1: Write Point-Slope Form

Using the point-slope form \(y - y_1 = m(x - x_1)\), where \(m=3\) is the slope and \((2,5)\) are the coordinates, the equation of the line becomes: \(y - 5 = 3(x - 2)\).

Step 2: Convert to Slope-Intercept form

By simplifying the point-slope form we get the slope-intercept form: Distribute the \(3\) on the right side of the equation to yield \(y - 5 = 3x - 6\), finally add \(5\) to both sides to get \(y = 3x - 1\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is key to solving problems related to lines in algebra. This form of an equation is expressed as \(y = mx + b\). Here, \(m\) stands for the slope, which tells us how steep the line is. The intercept \(b\) is where the line crosses the y-axis.
By familiarizing ourselves with this layout, it becomes easier to identify the characteristics of a line just by looking at its equation. For example:

• \(y = 2x + 3\) indicates a line with a slope of 2, which rises 2 units up for every 1 unit it moves to the right.
• The line also crosses the y-axis at the point \((0, 3)\).

In our exercise, converting from point-slope form to slope-intercept form helps to clearly see these elements and understand the line's behavior visually and mathematically.

Equation of a Line

An equation of a line represents all the points that lie on that line. It forms the basis of understanding how different points, slopes, and intercepts come together in a coordinate plane.
There are different forms for expressing these equations, among which the point-slope and slope-intercept forms are the most common.

• Point-Slope Form: \(y - y_1 = m(x - x_1)\)
• Slope-Intercept Form: \(y = mx + b\)

Knowing how to convert between these forms is a critical skill in algebra. Each form reveals different aspects of the line, making it easier to solve various problems depending on the given initial information. Whether a problem provides a point and a slope or simply requests the y-intercept, understanding the equation of a line enables comprehensive analysis and comprehension.

Algebra Concepts

Algebra simplifies solving problems by providing structured methods to describe and analyze mathematical relationships. One of the powerful concepts is using equations to understand the relationships between variables. When dealing with lines, we specifically look at:

• The slope, which is a measure of the steepness and direction.
• The y-intercept, which is where the line crosses the vertical axis.
• Coordinates, representing specific points on the graph.

Applying algebraic techniques allows us to transform complex statements into manageable equations, such as turning a point-slope form into a slope-intercept form. This not only simplifies calculations but also enhances our ability to predict and visualize the behavior of lines within a coordinate plane.