Problem 28 - 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. $$ (-2,-1),(8,8) $$

Step-by-Step Solution

Answer

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

Step 1: Calculate the Slope

The slope 'm' of a line through points \((-2,-1)\) and \( (8,8) \) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). If we substitute \(x_1 = -2\), \(y_1 = -1\), \(x_2 = 8\), \(y_2 = 8\), we get, the slope \(m = \frac{8 - (-1)}{8 - (-2)} = 1\).

Step 2: Calculate the y-intercept

Now, the formula of the line is mostly \(y = mx + c\) where 'c' is the y-intercept. We already have the slope 'm' as 1. We can substitute one of the given points, say \((-2,-1)\), and the 'm' value into the line equation to solve for 'c'. To do this, replace x with -2, y with -1, and m with 1. Doing this leads to \(-1 = 1*(-2) + c\). Solving this equation for 'c' produces c = 1.

Step 3: Write the equation

Now that we have 'm' = 1 and 'c' = 1, substitute these values into the line's equation \(y = mx + c\) to get the final equation of the line in slope-intercept form as \(y = x + 1\).

Key Concepts

Slope Calculation

Understanding how to calculate the slope is essential when dealing with linear equations. The slope indicates the steepness and direction of a line on a graph.
To find the slope, you need two points on a line. Let's denote these points as \( (x_1, y_1) \) and \( (x_2, y_2) \). The slope \(m\) is then calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

For our exercise with points \( (-2,-1) \) and \( (8,8) \), we plug these into the formula to get \(m = \frac{8 - (-1)}{8 - (-2)} = 1\). This result tells us that for every unit you move to the right on the x-axis, you'll move one unit up on the y-axis. Simple right? The slope of \(1\) indicates a line rising at a 45-degree angle, illustrating equal vertical and horizontal changes on the graph.

Graphing Linear Equations

Graphing linear equations involves plotting points and drawing a straight line through them. It's a visual way to represent the solution of an equation.

Let's use the slope calculation we've already done. With a slope of \(1\), and our points \( (-2,-1) \) and \( (8,8) \), we plot these exact points on graph paper or a digital graphing tool. Start at one point, use the slope to determine the next point, and connect your points with a straight line. Once you draw the line, double-check that it passes through both points to ensure accuracy.

Tips for Accurate Graphing

• Use a ruler or a straight edge for drawing the line.
• Extend the line across the graph to show it continues indefinitely.
• Always label your axes and points for clarity.

Y-Intercept Calculation

The y-intercept is where a line crosses the y-axis on a graph. Calculating it is a breeze once you know the slope and at least one point on the line.

With the slope \(m\) and a point from our exercise, \( (-2,-1) \), we plug these values into the slope-intercept formula \(y = mx + c\) to find the y-intercept \(c\). After substituting \(m = 1\), \(x = -2\), and \(y = -1\), we get \( -1 = 1\times (-2) + c \). When we solve for \(c\), we discover the y-intercept is \(1\).

This means that our line crosses the y-axis at the point \( (0,1) \). It’s an important value that, along with the slope, completely defines our line on a graph.

Writing Equations of Lines

Putting it all together, writing the equation of a line requires both the slope and the y-intercept. We've calculated a slope \(m=1\) and a y-intercept \(c=1\) for our line.

Using the slope-intercept form \(y = mx + c\), we replace \(m\) with \(1\), and \(c\) with \(1\) to get the final equation \(y = x + 1\). This equation represents every single point on the line, demonstrating the power of algebra in unifying countless points with one simple expression.

Remember:

• The coefficient of \(x\) in \(y = mx + c\) is the slope.
• The constant term \(c\) is the y-intercept.
• The slope-intercept form makes graphing and understanding lines incredibly straightforward.