Problem 30 - Algebra 1

Question

Graph the points and draw a line through them. Write an equation in slope- intercept form of the line that passes through the points. $$ (2,4),(1,-2) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (2, 4) and (1, -2) in slope-intercept form is \(y = 6x - 8\).

Step 1: Calculate the Slope

The slope \(m\) between points (2, 4) and (1, -2) can be calculated using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). So, \(m = (-2 - 4) / (1 - 2) = 6\).

Step 2: Determine the Y-intercept

To calculate the y-intercept \(c\), substitute the slope \(m\) and one of the points (take (2, 4) for instance) into the equation \(y = mx + c\). On substituting, we get \(4 = 6 * 2 + c\). Solving this will yield \(c = -8\).

Step 3: Write the Equation in Slope-intercept form

Now we have the slope \(m = 6\) and y-intercept \(c = -8\). Substituting these in the equation \(y = mx + c\), we get \(y = 6x - 8\). This is the equation of the line in slope-intercept form.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra that involves plotting points on a coordinate plane and connecting them to form a straight line. To graph a linear equation, you need to understand two things: the slope of the line and its y-intercept. The slope tells us how steep the line is, and the y-intercept indicates where the line crosses the y-axis.

Start by plotting the given points on the graph. For the points \(2,4)\) and \(1,-2)\), place a dot at each respective location on the Cartesian plane. Once the points are plotted, draw a straight line through them to visually see the equation's graph. The line graphed from our exercise represents the equation derived from these two points.

Calculating Slope

The slope of a line measures its steepness and direction. It is calculated by finding the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The formula for calculating slope \(m\) between any two points \( (x_1, y_1)\) and \( (x_2, y_2) \)) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

For our exercise's points \( (2, 4)\) and \( (1, -2)\), the slope is \(m = \frac{-2 - 4}{1 - 2} = 6\). A positive slope means the line rises from left to right, while a negative slope indicates the line falls. In this case, the slope 6 suggests that for each unit we move to the right, the line goes up by 6 units.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. Algebraically, it’s the value of \(y\) when \(x = 0\). When writing the equation of a line in slope-intercept form, which is \(y = mx + c\), \(c\) represents the y-intercept.

For the given exercise, after finding the slope, we use one of the points to solve for the y-intercept. Substituting the slope \(6\) and the point \(2, 4\) into the equation, we get \(4 = 6 \times 2 + c\). Solving for \(c\), we find that the y-intercept \(c = -8\). This y-intercept tells us that the line will cross the y-axis at \(y = -8\), which is a crucial starting point for graphing the line and understanding the equation's behavior.