Problem 6 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given points. $$ (-1,1) \text { and }(2,5) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (-1,1) and (2,5) is \(y = \frac{4}{3}x + \frac{7}{3}\).

Step 1: Find the Slope

The formula for finding the slope between two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) is given by: \(m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). Substitute the given points \((-1,1)\) and \((2,5)\) to find the slope. So, \(m = \frac{5-1}{2-(-1)} = \frac{4}{3}\).

Step 2: Find the Intercept

To find the y-intercept, substitute the slope and one of the points into the slope-intercept form. Let's use the point \((-1,1)\). Plugging these values into the equation gives: \(1 = \frac{4}{3}* -1 + b\). Solving for \(b\) gives \(b = \frac{7}{3}\).

Step 3: Write the Equation in Slope-Intercept Form

Finally, substitute the values for \(m\) and \(b\) back into the slope-intercept equation to get the desired equation in slope-intercept form: \(y = \frac{4}{3}x + \frac{7}{3}\).

Key Concepts

Equation of a Line

The equation of a line is a fundamental concept in algebra that helps us describe how two variables are related on a coordinate plane. The most common form of a linear equation is the slope-intercept form, written as:
\[ y = mx + b \] Here, \(y\) represents the dependent variable, \(x\) is the independent variable, \(m\) is the slope of the line, and \(b\) is the y-intercept.

• The slope \(m\) shows the steepness or incline of the line.
• The y-intercept \(b\) indicates where the line crosses the y-axis.

When given two points, you can use this information to construct the equation of the line by finding the slope and intercept. This form allows you to quickly visualize and understand how changes in \(x\) affect \(y\). Whether solving a simple algebra problem or plotting data for analysis, mastering the equation of a line is essential.

Finding Slope

Finding the slope of a line is a crucial step in creating its equation. The slope tells us how much \(y\) changes for a given change in \(x\). To find the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula calculates the 'rise' (change in \(y\)) over the 'run' (change in \(x\)).
In the problem, using points \((-1,1)\) and \((2,5)\), we compute:

• \(y_2 - y_1 = 5 - 1 = 4\)
• \(x_2 - x_1 = 2 - (-1) = 3\)
• Therefore, \(m = \frac{4}{3}\)

A positive slope like \(\frac{4}{3}\) indicates the line rises as \(x\) increases, providing insight into the relationship between the variables involved.

Y-intercept

The y-intercept is where the line crosses the y-axis, and it plays a key role in defining the line's equation. To find the y-intercept \(b\), use the slope-intercept formula after finding the slope:
Start with:\[ y = mx + b \]Substitute the slope and one point's coordinates into the equation.
For example, with point \((-1, 1)\) and slope \(\frac{4}{3}\):

• Set \(1 = \frac{4}{3}(-1) + b\)
• This simplifies to \(1 = -\frac{4}{3} + b\)
• Solving gives \(b = \frac{7}{3}\)

Understanding the y-intercept allows you to know exactly where the line meets the y-axis, giving a clearer picture of its position on the graph even before drawing it. Therefore, with both \(m\) and \(b\), the line equation becomes tangible and complete: \[ y = \frac{4}{3}x + \frac{7}{3} \]