Problem 24 - Elementary Algebra

Question

For the following problems, graph the equations. $$ \frac{1}{2} x+0 y=-1 $$

Step-by-Step Solution

Answer

Answer: The slope of the given equation is $$\frac{1}{2}$$, and the y-intercept is 0. To graph the equation, first plot the y-intercept point (0,0). Then, using the slope, move 2 units to the right and 1 unit up from the y-intercept to find another point on the graph (2,1). Connect these two points with a straight line to represent the graph of this equation.

Step 1: Identify the Slope and Y-intercept

The given equation is: $$ \frac{1}{2} x+0 y=-1 $$ This equation is already in the slope-intercept form, which is: $$ y = mx + b $$ where m is the slope and b is the y-intercept. In our case, the slope (m) is $$\frac{1}{2}$$ and the y-intercept (b) is 0, since: $$ y= -\frac{1}{2} x + 0 $$

Step 2: Plot the Y-Intercept

The y-intercept is the point where the graph intersects the y-axis. Since the y-intercept in our equation is 0, the point is (0,0). Start by plotting this point on the graph.

Step 3: Use the Slope to Find Another Point

The slope is a measure of how steep the line is. Since the slope (m) is $$\frac{1}{2}$$, this means that for every 2 units we move horizontally (to the right), we move 1 unit vertically (up). Starting from the y-intercept (0,0), move 2 units to the right and 1 unit up to find another point on our graph. This point is (2, 1).

Step 4: Draw the Line

Now we have two points (0,0) and (2,1). Connect these two points with a straight line, making sure it extends beyond both points. This line represents the graph of the equation $$-\frac{1}{2}x+0y=-1$$. Now the student can visualize the graph of the given equation and better understand the relationship between the slope, the y-intercept, and the graph.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when graphing linear equations. This form is written as: ewline $$ y = mx + b $$ where is the dependent variable, usually representing the vertical position on the graph, is the slope of the line, which determines its steepness, and is the y-intercept, the point at which the line crosses the y-axis.

In our exercise, the equation ewline $$ \frac{1}{2}x + 0 y = -1 $$ is already in the slope-intercept form. By rearranging it to follow the template, we recognize that equals to ewline $$ \frac{1}{2} $$ and the equals to 0. The slope ewline $$ \frac{1}{2} $$ indicates that for every one unit increase in , there's a half unit increase in , hence the 'slope' signifies the direction and steepness of our line in question.

Through slope-intercept form, we immediately know two crucial pieces of information: where to start plotting our graph (the y-intercept) and how to proceed with drawing the line (direction and steepness defined by the slope).

Plotting Y-Intercept

The y-intercept is an essential starting point when graphing linear equations. It's the coordinate on the graph where our line will cross the y-axis. In mathematical terms, it's the value of ewline when ewline is 0.

In the provided exercise, we've identified that our y-intercept b is 0. So, the coordinate of the y-intercept is (0,0). Plotting this point gives us a reference from which to apply the slope and determine the direction of the line. To plot the y-intercept:

• Locate the origin (0,0) on the graph.
• Place a point or make a mark at this location.
• This point will serve as an anchor for drawing our line.

Remember, the y-intercept is not just a point but a signal to where your line will 'intersect' with the 'y' axis, hence the term y-intercept. From here, you can use the slope to find other points and draw the line with precision.

Slope of a Line

The slope of a line is a measure of how fast ewline changes as ewline changes. It's represented by the letter and is often thought of as 'rise over run'. This essentially means how much ewline (the rise) goes up or down for every unit of horizontal change in ewline (the run).

The slope given in our exercise is ewline $$ \frac{1}{2} $$ which means for every 2 units the line runs to the right (along the x-axis), it rises 1 unit (along the y-axis). To use this information:

• Starting at the y-intercept, which is at (0,0) for this problem, you move horizontally to the right on the graph.
• For every 2 units moved to the right, you will move up 1 unit to find the next point.
• Plot the second point (2,1) after moving according to the slope.
• Connect this point back to the y-intercept to extend the line in both directions.

It is through the slope that we determine the orientation of the line on the graph. A positive slope like the one we have indicates the line goes upwards from left to right. Conversely, a negative slope would mean the line goes downwards when moving from left to right.