Problem 27 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given points. $$ (-1,-2) \text { and }(2,6) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (-1,-2) and (2,6) is \(y = \frac {8} {3}x + \frac {2} {3}\).

Step 1: Calculate the Slope

The formula for the slope 'm' between two points (x1,y1) and (x2,y2) is given by \[ m = \frac {y_2-y_1} {x_2-x_1} .\] Using the given points in the question (-1,-2) and (2,6), substituting \(x_1 = -1\), \(y_1 = -2\), \(x_2 = 2\), \(y_2 = 6\) into the formula, we get \[ m = \frac {6-(-2)} {2-(-1)} = \frac {8} {3} .\] So the slope of the line is \(\frac {8} {3}\).

Step 2: Find the Y-Intercept

Now, using the slope-intercept formula \(y = mx + c\), we can substitute the values of one of the points and the slope to solve for 'c'. Let's use the point (-1,-2). We have \[ -2 = \frac {8} {3} (-1) + c \]. Solving for 'c', we find that \(c = -2 + \frac {8} {3} = \frac {2} {3}\).

Step 3: Write the Equation of the Line

Now that we have the slope \(m = \frac {8} {3}\) and the y-intercept \(c = \frac {2} {3}\), we substitute these values into the slope-intercept formula \(y = mx + c\) to get the equation of the line. The final equation of the line is \[ y = \frac {8} {3}x + \frac {2} {3} .\]

Key Concepts

Calculating Slope

Understanding how to calculate the slope is crucial when you're dealing with linear equations. The slope measures how steep a line is and is usually represented by the letter 'm'. To find the slope between two points, you use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \.\]
For example, if you're given two points A \( (-1, -2) \) and B \( (2, 6) \),
you subtract the y-coordinate of point A from point B and divide by the subtraction of the x-coordinate of point A from point B, like this:\[ m = \frac{6 - (-2)}{2 - (-1)} = \frac{8}{3} \]
Notice how we turn subtraction of a negative number into addition. The result is the slope, which in this example, is \( \frac{8}{3} \), a positive fraction indicating the line inclines as it moves from left to right.

Y-intercept

The y-intercept is where the line crosses the y-axis. It's a vital part of the line's equation and is represented by the letter 'c' in the slope-intercept form \( y = mx + c \). Finding the y-intercept means figuring out what value y will have when x is zero. To solve for the y-intercept, rearrange the slope-intercept equation to isolate 'c'. For the line passing through \( (-1, -2) \) and \( (2, 6) \) with a slope of \( \frac{8}{3} \), we plug the coordinates of one of the points into this formula:
\[ c = y - mx \].
In this case, using point A \( (-1, -2) \) results in:\[ c = -2 - \frac{8}{3}(-1) = -2 + \frac{8}{3} = \frac{2}{3} \].
So, the y-intercept of our line is \( \frac{2}{3} \).

Equation of a Line

With both the slope and y-intercept calculated, we can now write the equation of the line. In slope-intercept form, the equation of any line is expressed as \( y = mx + c \), where m is the slope, and c is the y-intercept. Given our slope \( \frac{8}{3} \) and y-intercept \( \frac{2}{3} \), the equation for the line through our two points is:\[ y = \frac{8}{3}x + \frac{2}{3} \].
This equation allows us to calculate the value of y for any corresponding value of x, meaning it represents all the points that line up straight through \( (-1, -2) \) and \( (2, 6) \) on a Cartesian plane.

Algebraic Concepts

Slope and y-intercept are part of fundamental algebraic concepts involved in analyzing and creating linear equations.
Algebra is the branch of mathematics concerning the study of rules of operations and the constructions arising from them. When dealing with equations of a line, you engage in algebraic manipulation: moving terms around, balancing equations, and solving for unknowns,
which ultimately allows you to graph lines or predict values within a particular domain. These algebraic skills are not only crucial for plotting linear equations but also foundational for more complex mathematical concepts later on in your education path. Each step taken to solve the slope-intercept form exercises reinforces your understanding of algebraic fundamentals.