Solving Systems Of Equations Graphically An Example With Shannon
Solving systems of equations is a fundamental concept in mathematics, and graphical methods offer a visual and intuitive approach to finding solutions. In this article, we will delve into the process of solving a system of equations graphically, using the specific example Shannon graphed:
7x - 4y = -8
y = (3/4)x - 3
We will explore each step in detail, from understanding the equations to plotting the lines and identifying the point of intersection, which represents the solution to the system. This comprehensive guide aims to provide a clear understanding of the graphical method, enabling you to confidently solve similar systems of equations.
Understanding the Equations
Before diving into the graphical solution, it's crucial to understand the nature of the equations we're dealing with. The given system consists of two linear equations:
- 7x - 4y = -8
- y = (3/4)x - 3
Linear equations represent straight lines when plotted on a coordinate plane. Each equation describes a relationship between the variables x and y. The solution to a system of linear equations is the point (x, y) that satisfies both equations simultaneously. Graphically, this solution corresponds to the point where the lines representing the equations intersect.
To effectively graph these equations, it's helpful to express them in slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept. The second equation is already in slope-intercept form, making it straightforward to graph. Let's transform the first equation into slope-intercept form:
7x - 4y = -8
Subtract 7x from both sides:
-4y = -7x - 8
Divide both sides by -4:
y = (7/4)x + 2
Now, both equations are in slope-intercept form:
- y = (7/4)x + 2
- y = (3/4)x - 3
This form provides valuable information about the lines. For the first equation, the slope is 7/4, and the y-intercept is 2. For the second equation, the slope is 3/4, and the y-intercept is -3. The slope indicates the steepness of the line, while the y-intercept is the point where the line crosses the y-axis. These pieces of information are essential for accurately graphing the lines.
Plotting the First Line: y = (7/4)x + 2
To plot the first line, y = (7/4)x + 2, we can use the slope-intercept form. The y-intercept is 2, which means the line passes through the point (0, 2). We can mark this point on the coordinate plane. The slope is 7/4, which means for every 4 units we move to the right on the x-axis, we move 7 units up on the y-axis. Starting from the y-intercept (0, 2), we can move 4 units to the right and 7 units up to find another point on the line. This brings us to the point (4, 9). Now, we can draw a straight line passing through these two points. To ensure accuracy, it's always a good idea to find and plot at least three points. Let's find one more point. If we move 4 units to the left from the y-intercept (0,2) and 7 units down, we find the point (-4, -5). Plotting this third point provides a visual check for our line's accuracy. After plotting these points, you can use a ruler or straight edge to draw a line that smoothly connects all three. This line represents all the possible solutions to the equation y = (7/4)x + 2.
Plotting the Second Line: y = (3/4)x - 3
Now, let's plot the second line, y = (3/4)x - 3. Again, we'll use the slope-intercept form. The y-intercept is -3, so the line passes through the point (0, -3). Mark this point on the coordinate plane. The slope is 3/4, which means for every 4 units we move to the right on the x-axis, we move 3 units up on the y-axis. Starting from the y-intercept (0, -3), move 4 units to the right and 3 units up. This gives us the point (4, 0). Plot this point. For an additional point, we can move another 4 units to the right and 3 units up from (4, 0), leading us to the point (8, 3). Plot this third point. Now, draw a straight line passing through all three points. As with the first line, plotting multiple points helps ensure the line's accuracy. This line represents all the possible solutions to the equation y = (3/4)x - 3. With both lines now graphed on the coordinate plane, we are ready to identify the solution to the system of equations.
Identifying the Point of Intersection
The point where the two lines intersect represents the solution to the system of equations. This point (x, y) satisfies both equations simultaneously. By visually inspecting the graph, we can estimate the coordinates of the point of intersection. It's important to note that graphical solutions may not always be exact, especially if the point of intersection falls between grid lines. In such cases, algebraic methods are often used to find a more precise solution. However, the graphical method provides a clear visual representation of the solution and is a valuable tool for understanding systems of equations.
In this specific example, by carefully examining the graph, we can see that the two lines intersect at the point (4, 0). This means that the solution to the system of equations is x = 4 and y = 0. To verify this solution, we can substitute these values into the original equations:
- 7x - 4y = -8 7(4) - 4(0) = 28 - 0 = 28 ≠-8
- y = (3/4)x - 3 0 = (3/4)(4) - 3 = 3 - 3 = 0
Upon substituting the values, we notice that the first equation is not satisfied by our graphical solution. It appears there was a slight error in reading the intersection point from the graph. Let's re-examine our equations and use an algebraic method to confirm the correct solution.
Algebraic Verification and Solution
To find the exact solution to the system of equations, we can use algebraic methods such as substitution or elimination. Let's use the substitution method, since the second equation is already solved for y:
- 7x - 4y = -8
- y = (3/4)x - 3
Substitute the expression for y from equation (2) into equation (1):
7x - 4((3/4)x - 3) = -8
Distribute the -4:
7x - 3x + 12 = -8
Combine like terms:
4x + 12 = -8
Subtract 12 from both sides:
4x = -20
Divide both sides by 4:
x = -5
Now that we have the value of x, we can substitute it back into equation (2) to find y:
y = (3/4)(-5) - 3 y = -15/4 - 12/4 y = -27/4
So, the exact solution to the system of equations is x = -5 and y = -27/4. This highlights the importance of algebraic methods for obtaining precise solutions, especially when graphical methods might lead to approximations.
Implications of the Solution
The solution x = -5 and y = -27/4 represents the coordinates of the point where the two lines intersect on the coordinate plane. This point is the only point that lies on both lines, thus satisfying both equations. In the context of real-world applications, systems of equations are used to model various scenarios, and the solution represents a specific set of values that meet the conditions of the problem. For instance, these equations could represent cost and revenue functions, and the solution would indicate the break-even point where costs equal revenue.
The graphical method, while providing a visual understanding, may not always yield precise solutions. As we saw in our initial attempt to read the intersection point from the graph, slight inaccuracies can occur. Therefore, it's crucial to verify the solution algebraically to ensure its accuracy. This combination of graphical and algebraic methods provides a robust approach to solving systems of equations.
Conclusion
Shannon's graphical solution to the system of equations 7x - 4y = -8 and y = (3/4)x - 3 illustrates the power of visual methods in understanding mathematical concepts. By plotting the lines and identifying the point of intersection, we gain a visual representation of the solution. However, it's essential to complement graphical methods with algebraic techniques to ensure the accuracy of the solution. In this case, the algebraic solution revealed the precise point of intersection to be (-5, -27/4), highlighting the importance of a comprehensive approach to problem-solving in mathematics. The combination of graphical and algebraic methods not only provides a solution but also enhances our understanding of the underlying principles of systems of equations.
