Problem 52 - Algebra 1

Question

Find the slope and the \(y\) -intercept of the graph of the equation. Then graph the equation. $$ 4 x+2 y=6 $$

Step-by-Step Solution

Answer

The slope of the line represented by the equation \(4x + 2y = 6\) is -2 and the y-intercept is 3. The graph of the equation is a line that crosses the y-axis at 3 and slopes downward from left to right.

Step 1: Convert to Slope-Intercept Form

The first step is to convert the given equation from standard form to slope-intercept form. The slope-intercept form is \(y = mx + b\), where m is the slope and b is the y-intercept. We can get this form by isolating y in the equation. So, the equation can be rewritten as: \(2y = -4x + 6\), and then divide every term by 2 to have y in terms of x. So we get \(y = -2x + 3\).

Step 2: Identify the Slope and y-Intercept

Now, we have the equation in slope-intercept form and we can easily identify the slope and the y-intercept. From the equation \(y = -2x + 3\), the number attached to x, which is -2, is the slope, and 3 is the y-intercept.

Step 3: Plot the Graph

Now we can plot the line using the slope and the y-intercept that we found. First, plot a point on the y-axis at 3, which is the y-intercept. Then, since the slope is -2 (or -2/1), from the y-intercept, we move down 2 units and right 1 unit to get the next point. We connect these points to represent the line.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing lines and analyzing their properties. This form is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) denotes the y-intercept.

The beauty of this form is in its directness; it gives you a clear visual guide of how a line behaves. The slope, \( m \), tells you how steep the line is and in which direction it tilts—upward for positive slopes, downward for negative ones. On the other hand, the y-intercept, \( b \), tells you where the line crosses the y-axis. This particular point is where the value of \( x \) is zero.

So, when you're given a linear equation, like in our original exercise \( 4x + 2y = 6 \), converting it to the slope-intercept form makes it more intuitive to understand and much easier to graph.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Calculated as the ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run), it's represented as \( m \) in the slope-intercept form \( y = mx + b \). A positive slope means that the line rises as it goes from left to right, while a negative slope indicates that the line falls.

In our example, after rearranging the equation to the slope-intercept form, we've identified the slope to be -2. This number means that for every step right (positive x direction), the line goes down by two steps. It's a crucial concept for graphing the line accurately and for understanding how one variable relates to another in the given equation.

Y-Intercept

The y-intercept of a line is a fundamental characteristic that represents the exact point where the line crosses the y-axis. This point is where the value of \( x \) is zero, and it can be read directly from the slope-intercept form of the line, as the \( b \) value.

Say you have a line with the equation \( y = -2x + 3 \). The y-intercept here is 3, which translates to the point \( (0, 3) \) on the graph. Starting at the y-intercept makes plotting the rest of the line straightforward, as you then use the slope to determine the line's direction and additional points. Remember, every linear equation will have a unique y-intercept, giving each line its own starting point on a graph.

Plotting Graphs

Plotting graphs of linear equations visually represents the relationship between two variables. The process begins with identifying the y-intercept and marking it on the graph. From there, you use the slope to find additional points.

For example, with a slope of -2, you move two units down and one unit to the right from the y-intercept. Repeat this step, and soon you'll have enough points to draw a straight line.

Drawing the Line

Once your points are plotted, take a ruler and connect them, extending the line through and beyond the points to ensure it covers the entire range of the graph. Voilà, you have successfully graphed a linear equation, which now serves as a visual tool for analyzing the relationship between the variables.