Problem 35 - Elementary Algebra

Question

For the following problems, graph the equations. $$ 4.1 x-6.6 y=15.5 $$

Step-by-Step Solution

Answer

Question: Determine and graph two points on the line represented by the equation \(4.1x - 6.6y = 15.5\). Answer: The x-intercept is the point (3.78, 0), and the y-intercept is the point (0, -2.35). To graph the line, plot these points and draw a straight line that passes through them.

Step 1: Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. $$4.1x - 6.6(0) = 15.5$$ $$4.1x = 15.5$$ Now, divide by 4.1 to get x. $$x = \frac{15.5}{4.1}$$ $$x = 3.78$$ The x-intercept is the point (3.78, 0).

Step 2: Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. $$4.1(0) - 6.6y = 15.5$$ $$- 6.6y = 15.5$$ Now, divide by -6.6 to get y. $$y = \frac{15.5}{-6.6}$$ $$y = -2.35$$ The y-intercept is the point (0, -2.35).

Step 3: Graph the points and draw the line

Plot the x-intercept (3.78, 0) and the y-intercept (0, -2.35) on the coordinate plane. Then, draw a straight line that passes through both points. This line represents the graph of the equation \(4.1x - 6.6y = 15.5\).

Key Concepts

X-Intercept

Understanding the x-intercept is crucial when graphing linear equations. It is the point where the line crosses the x-axis on the coordinate plane. To find this point, you simply need to set the value of y to zero and solve the equation for x.

In the exercise, the equation is set as follows:
\(4.1x - 6.6(0) = 15.5\). By solving \(4.1x = 15.5\), we calculate the x-coordinate of the intercept as \(\frac{15.5}{4.1}\), which results in \(x = 3.78\). Hence, the x-intercept is the point (3.78, 0). Marking this point on a graph allows you to start plotting the linear equation.

Y-Intercept

The y-intercept also plays a key role in graphing linear functions; it's where the line meets the y-axis. To determine the y-intercept, set x to zero and solve for y. This is reflected in the step-by-step solution:
\(4.1(0) - 6.6y = 15.5\).
Once we have \( -6.6y = 15.5 \), we can find the y-coordinate by dividing both sides by -6.6, resulting in \(y = \frac{15.5}{-6.6}\), which simplifies to \(y = -2.35\). Thus, the y-intercept is (0, -2.35). Identifying the y-intercept helps to secure another reference point on the graph, leading to a more accurate representation of the linear equation.

Coordinate Plane

The coordinate plane is the canvas for algebraic graphing. It is a two-dimensional surface formed by two number lines intersecting at a right angle. The horizontal line is called the x-axis, while the vertical line is the y-axis. Their intersection is known as the origin, with coordinates (0,0).

When graphing linear equations, it's useful to remember that every point on the plane is expressed as (x,y), where x and y represent the values on their respective axes. Plotting the x-intercept and y-intercept on the coordinate plane provides two crucial points that define the straight line representing a linear equation.

Algebraic Graphing

Algebraic graphing is the process of representing equations or functions visually on a coordinate plane. Linear equations, such as the one given in our exercise, form straight lines when graphed. To do this effectively, you start with the intercepts.

After calculating the intercepts (3.78, 0) for the x-axis and (0, -2.35) for the y-axis, you plot these points and draw a line through them, extending it across the plane. This line represents all the solutions to the equation \(4.1x - 6.6y = 15.5\).

Remember that every point on this line is a solution to the equation. This graph now serves as a visual aid, making it easier to understand the relationship between x and y as dictated by the given linear equation.