Problem 37 - Algebra 1

Question

Graph the line that has the given intercepts. $$ \begin{array}{l} x \text { -intercept: }-7 \\ y \text { -intercept: }-3 \end{array} $$

Step-by-Step Solution

Answer

The required line passes through the points (-7,0) and (0,-3).

Step 1: Plot the x-intercept

The x-intercept is -7, which means the line crosses the x-axis at -7. This results in the point (-7,0). Plot this point on the graph.

Step 2: Plot the y-intercept

The y-intercept is -3. This means the line crosses the y-axis at -3. Consequently the point of intersection is (0,-3). Plot this point on the graph as well.

Step 3: Draw the Line

Now that the points (-7,0) and (0,-3) have been plotted on the graph, draw a straight line through these two points. This line represents the equation of the line with the given intercepts.

Key Concepts

X-Intercept

Understanding the x-intercept of a line is fundamental when learning about graphing. This is the point where the line crosses the x-axis. It's found by looking at the graph and identifying the spot where the line hits the x-axis, thus making the y-coordinate zero. For instance, if you have an x-intercept of -7, as mentioned in our example, this means that the corresponding point on the graph is (-7, 0). To visualize this, you simply find -7 on the x-axis (our horizontal axis), and mark a point where this value intersects with the axis. Remember, since this is an x-intercept, the y-value will always be zero. In exercises, you can improve your comprehension by always starting with plotting the x-intercept, as it lays the foundation for your line.

Y-Intercept

In contrast to x-intercepts, y-intercepts are where a line crosses the y-axis, and thereby have an x-coordinate of zero. Taking our initial problem as an example, the y-intercept given is -3. To plot this on a graph, you locate -3 on the y-axis (our vertical axis), and mark the point (0, -3). By plotting this point, it signals where the line will cross the y-axis. It's important to understand that every point on the y-axis will have an x-coordinate of zero, just as every point on the x-axis will have a y-coordinate of zero for the x-intercept. Plotting the y-intercept is equally significant because it helps to define the steepness and direction of the line on the graph.

Plotting Points on a Graph

Graphing a line requires plotting points accurately. Each point is defined by a pair of numbers called coordinates, which represent its position on the x-axis (horizontal) and the y-axis (vertical). To plot a point, such as (-7, 0) or (0, -3) from our exercise, start by locating the x-value on the x-axis then move vertically to reach the y-value. Plotting multiple points and connecting them with a straight edge can bring the abstract concept of a line into tangible form. Frequent practice in plotting points will increase your ability to visualize mathematical concepts and translate equations onto a graph seamlessly.

Equation of a Line

Every straight line can be represented by an equation, and understanding it is crucial for graphing. The most straightforward form of a line's equation is the slope-intercept form: \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. In scenarios where you have intercepts but not the explicit equation, such as the initial problem provided, you can determine the line's equation by using the intercepts. Using the points (-7, 0) and (0, -3) from the exercise, you would find the change in y over the change in x to determine the slope, and you already have the y-intercept. With practice, you'll get better at identifying these aspects of a line, and transitioning from the visual graph to the algebraic equation will become second nature.