Problem 46 - Algebra 1

Question

Write the point-slope form of the equation of the line that passes through the point and has the given slope. Then rewrite the equation in slope-intercept form. $$ (4,-2), m=\frac{1}{4} $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 2 = \frac{1}{4}x - 1\) and in slope-intercept form it is \(y = \frac{1}{4}x - 3\).

Step 1: Writing the Equation in Point-Slope Form

Given the point (4,-2) and slope \(m=\frac{1}{4}\), let's substitute these values into the point-slope form equation. So we get \(y - (-2) = \frac{1}{4} (x - 4)\).

Step 2: Simplifying the Equation

Simplify the equation by removing the parenthesis and reducing any possible terms. So the equation becomes \(y + 2 = \frac{1}{4}x - 1\).

Step 3: Rewrite the Equation in Slope-Intercept Form

To transform the equation to slope-intercept form \(y = mx + c\), we just need to isolate \(y\) on one side. Subtract 2 from both sides to get \(y = \frac{1}{4}x - 3\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line. This form makes it easy to see both the slope and the y-intercept, providing a quick understanding of how the line behaves on a graph.

The general formula for slope-intercept form is \[y = mx + b\] where:

• y is the dependent variable.
• x is the independent variable.
• m represents the slope of the line, indicating the steepness and direction.
• b is the y-intercept, the point where the line crosses the y-axis.

To convert an equation into slope-intercept form, you need to rearrange the equation so that it resembles \(y = mx + b\). This often involves isolating \(y\) and simplifying the equation if necessary.

In the solution above, the equation \(y = \frac{1}{4}x - 3\) is in slope-intercept form, showing that for every increase of 1 in \(x\), \(y\) increases by \(\frac{1}{4}\), while the line crosses the y-axis at -3.

Linear Equations

Linear equations are mathematical statements that graph as straight lines. They represent a constant relationship between two variables, typically \(x\) and \(y\). Such equations take various forms, but their defining characteristic is a constant rate of change, or slope.

Linear equations appear in several forms:

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

Each form has its unique advantages depending on the information available. In practical situations, the slope-intercept form is often favored for its simplicity when graphing a line. Linear equations model many real-world phenomena, such as computing velocity in physics or predicting costs in economics.

In the above solution, the line passing through the point (4,-2) with a slope of \(\frac{1}{4}\) is represented by a linear equation. The switch from point-slope form to slope-intercept form, \(y = \frac{1}{4}x - 3\), reveals the direct impact of changes in \(x\) on changes in \(y\).

Equation of a Line

The equation of a line in mathematics describes the infinite points along a straight path. It provides a way to predict where the line will be at any given value of \(x\). There are several forms of line equations, each suited to different scenarios.

Different line equations include:

• Slope-Intercept Form: Highlights slope and y-intercept.
• Point-Slope Form: Useful when a point on the line and the slope are known.
• Standard Form: Often used for calculations involving multiple lines.

In problems like the one provided, knowing just one point and the slope helps you establish the line's equation using the point-slope form. From there, converting to slope-intercept form offers a clear perspective on how the line behaves.
The point-slope form \(y - y_1 = m(x - x_1)\) is a starting point when you know a single point and the slope. Specifically, in our given solution, it starts with \(y + 2 = \frac{1}{4}x - 1\) and simplifies to tell us precisely how the line behaves across the coordinate plane in the familiar form \(y = \frac{1}{4}x - 3\).