Problem 24 - Algebra 1

Question

Graph the equation. $$ y=x+5 $$

Step-by-Step Solution

Answer

The graph of the equation \(y=x+5\) is a straight line passing through the points \(0,5\) and \(1,6\). It has a slope of \(1\).

Step 1: Identifying the equation

The equation to be graphed is \(y=x+5\). This equation is of form \(y=mx+c\) where \(m\) is the slope and \(c\) is the y intercept.

Step 2: Find the y-intercept

Looking at the equation, it can be observed that the \(y\) intercept is \(5\). This means the line crosses the y axis at point \(0,5\).

Step 3: Find another point

Using the slope (which is \(1\) in this case), move one unit up and one unit to the right from the y-intercept. This point is \(1,6\).

Step 4: Plotting the graph

Plot the points \(0,5\) and \(1,6\) on the graph. Then, draw a straight line passing through these points. The line should be extending beyond these points as the line represented by the equation is infinite.

Key Concepts

Understanding the Y-Intercept

The y-intercept of a linear equation is simply the point where the line represented by the equation crosses the y-axis. Imagine the y-axis as a vertical line on a graph; it's where the x-coordinate is always zero. The y-intercept is described by the coordinates (0, c) where 'c' is a constant.

To find the y-intercept of the equation \(y = x + 5\), we look for the value of y when x equals zero. By substituting x with 0, we quickly determine that the y-intercept is 5, or in coordinate form, (0, 5). This point is one of the first and most crucial steps when plotting a graph because it gives us a definitive starting position on the graph where we can anchor our line.

Deciphering the Slope

The slope of a line on a coordinate plane describes how steep the line is - whether it rises or falls as it moves from left to right. The slope is often denoted as 'm' and is calculated by the rise over the run, meaning the change in y over the change in x.

In the equation \(y = x + 5\), the slope is the coefficient of x, which in this case is 1. This means for every one unit increase in x, y increases by one unit, which is a moderate and even ascent. It's a positive slope, indicating the line rises as we move to the right. Recognizing the slope allows us to find another point on the line; by starting at the y-intercept and moving according to slope (in this case up 1 and right 1), we land on the second point necessary for plotting, which is (1, 6).

Plotting on a Coordinate Plane

Putting it all together involves coordinate plane plotting, which is where the visual aspect of algebra comes to life. A coordinate plane has two axes: the horizontal x-axis and the vertical y-axis, allowing us to map points according to their x and y coordinates.

To graph the equation \(y = x + 5\), we first plot the y-intercept (0, 5). Next, using the slope of 1, we move to our next point (1, 6). These two points are like dots we've placed on our mathematically-induced map. By drawing a straight line through these points and extending it on both sides, we have successfully graphed our linear equation. This line represents all the points that satisfy the equation, forming a visual representation of the relationship between x and y values in the equation.