Problem 19 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form. $$(-3,2), m=\frac{1}{3}$$

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac{1}{3}x + 3\)

Step 1: Identify the given slope and the point

The given slope \(m\) is \(\frac{1}{3}\) and the given point is (-3,2).

Step 2: Substitute the slope, and the coordinates of the point into the formula

Plugging our values into the formula \(b = y - mx\) for the y-intercept gives us \(b = 2 - (-3)*\(\frac{1}{3}\)

Step 3: Solve for the y-intercept

Solving the equation gives us a y-intercept \(b = 2 - (-1) = 2 + 1 = 3\)

Step 4: Write the final equation of the line

Now that we have \(m\) and \(b\), we can write the equation of the line in slope-intercept form. It is \(y = \frac{1}{3}x + 3\)

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is a foundational concept in algebra. It is generally expressed as
\( y = mx + b \),
where \( m \) represents the slope of the line, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis. Here's why the slope-intercept form is so useful:

• It provides a quick way to graph a linear equation since it directly shows the slope and y-intercept.
• The format is standardized, making it easier to compare different linear equations.
• It simplifies the process of finding other characteristics of the line, such as intercepts and slope.

Let's apply the slope-intercept form to the exercise given, where the slope \( m \) is \( \frac{1}{3} \) and the y-intercept \( b \) is 3. Using the formula, the final equation for the line that passes through the point \( (-3,2) \) with a slope of \( \frac{1}{3} \) is \( y = \frac{1}{3}x + 3 \). This equation gives a clear, visual representation of the line when plotted on a graph.

Linear Equations

Linear equations form a straight line when graphed on a coordinate plane, and they typically involve two variables, x and y. The equations can appear in several forms, including the standard form (Ax + By = C) and the slope-intercept form \( y = mx + b \).

• Linear equations represent a constant rate of change, which is the slope of the line.
• They can model real-world situations, such as predicting trends and understanding relationships between two variables.

For the given exercise, after solving for the y-intercept, we have represented all necessary information as a linear equation in slope-intercept form. Furthermore, the process of converting points and a slope into a complete equation is central to understanding how linear functions behave.

Writing Equations from a Point and Slope

To write the equation of a line given a point and a slope, one can use the point-slope form of the equation: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point the line passes through.
Here's how to do it:

• Substitute the slope \( m \) and the coordinates of the point into the point-slope formula.
• Rearrange the equation to solve for \( y \) to get it into the slope-intercept form.

Using our exercise as an example, with the point \( (-3,2) \) and slope \( \frac{1}{3} \), a key step is to calculate the y-intercept using the formula mentioned in the steps. Once we determine that the y-intercept \( b \) is 3, we can write the equation in the slope-intercept form highlighted above. It's essential to remember these concepts when working with linear equations, as they form the basis for understanding more complex algebraic concepts.