Solving Linear Equations 3x - 4y = 7 And X - 4y = 3 Graphically
In this article, we will delve into the process of solving a system of linear equations and graphically illustrating the feasible region. Linear equations are fundamental in mathematics and have wide applications in various fields such as economics, engineering, and computer science. Understanding how to solve these equations and visualize their solutions is crucial for problem-solving and decision-making.
Understanding Linear Equations
Before we dive into solving the specific equations, let's briefly discuss what linear equations are. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variables are raised to the power of one, and there are no products or other non-linear functions of the variables. A system of linear equations is a set of two or more linear equations involving the same variables.
The general form of a linear equation in two variables (x and y) is:
Ax + By = C
Where A, B, and C are constants.
Solving a system of linear equations means finding the values of the variables that satisfy all the equations simultaneously. There are several methods to solve linear equations, including substitution, elimination, and graphical methods.
The Given Equations
We are given the following system of linear equations:
- 3x - 4y = 7
- x - 4y = 3
Our goal is to find the values of x and y that satisfy both equations and then graphically represent the feasible region defined by these equations.
Solving the Equations
Let's use the elimination method to solve the given system of equations. The elimination method involves manipulating the equations so that the coefficients of one of the variables are the same (or opposites) in both equations. Then, we can add or subtract the equations to eliminate that variable.
In this case, we notice that the coefficient of y is -4 in both equations. Therefore, we can subtract the second equation from the first equation to eliminate y:
(3x - 4y) - (x - 4y) = 7 - 3
Simplifying the equation, we get:
3x - 4y - x + 4y = 4
2x = 4
Dividing both sides by 2, we find:
x = 2
Now that we have the value of x, we can substitute it into either of the original equations to find the value of y. Let's substitute x = 2 into the second equation:
2 - 4y = 3
Subtracting 2 from both sides, we get:
-4y = 1
Dividing both sides by -4, we find:
y = -1/4
Therefore, the solution to the system of equations is x = 2 and y = -1/4.
Graphical Representation
To graphically represent the equations, we need to plot them on a coordinate plane. Each linear equation represents a straight line. The solution to the system of equations is the point where the two lines intersect.
Let's rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
- 3x - 4y = 7
-4y = -3x + 7
y = (3/4)x - 7/4 2. x - 4y = 3
-4y = -x + 3
y = (1/4)x - 3/4
Now we can plot these lines on a graph. For the first equation, the slope is 3/4 and the y-intercept is -7/4. For the second equation, the slope is 1/4 and the y-intercept is -3/4.
The point of intersection of the two lines represents the solution to the system of equations, which we found to be (2, -1/4). This point lies on both lines and satisfies both equations.
Feasible Region
The feasible region is the set of all points that satisfy the inequalities associated with the linear equations. In this case, we have equations, not inequalities. So, if we consider these equations as boundaries, there is no feasible region in the traditional sense (an area). However, if we were to consider inequalities like:
- 3x - 4y ≤ 7
- x - 4y ≤ 3
Then, the feasible region would be the area that satisfies both inequalities. This area would be the intersection of the regions defined by each inequality. To determine which side of the line to shade for each inequality, we can test a point (e.g., (0,0)) in the original inequality.
For the first inequality, 3(0) - 4(0) ≤ 7, which simplifies to 0 ≤ 7, which is true. So, we would shade the region on the same side of the line as the point (0,0).
For the second inequality, (0) - 4(0) ≤ 3, which simplifies to 0 ≤ 3, which is also true. So, we would shade the region on the same side of the line as the point (0,0).
The feasible region would be the area where the shaded regions of both inequalities overlap.
If the inequalities were:
- 3x - 4y ≥ 7
- x - 4y ≥ 3
Then, we would shade the opposite side of the lines because (0,0) would not satisfy the inequalities.
If we had a mix of ≥ and ≤, the feasible region would be the intersection of the regions defined by the different inequalities, shading the appropriate side for each one.
Example with Inequalities
Let's consider the following inequalities to illustrate the concept of a feasible region:
- x + y ≤ 5
- x ≥ 0
- y ≥ 0
To find the feasible region, we first graph the lines corresponding to the inequalities:
- x + y = 5
- x = 0
- y = 0
The line x + y = 5 has intercepts (5,0) and (0,5). The line x = 0 is the y-axis, and the line y = 0 is the x-axis.
Now we need to determine which side of each line to shade. For the inequality x + y ≤ 5, we can test the point (0,0): 0 + 0 ≤ 5, which is true, so we shade the region on the same side of the line as (0,0).
For the inequality x ≥ 0, we shade the region to the right of the y-axis.
For the inequality y ≥ 0, we shade the region above the x-axis.
The feasible region is the triangular area bounded by the lines x + y = 5, x = 0, and y = 0. This region includes all points (x, y) that satisfy all three inequalities simultaneously.
Importance of Graphical Representation
Graphical representation is a powerful tool for visualizing solutions to linear equations and inequalities. It allows us to see the relationship between the equations and the variables, and it provides a clear picture of the feasible region. This is particularly useful in optimization problems, where we want to find the maximum or minimum value of a function subject to certain constraints.
By graphically representing the constraints (inequalities), we can identify the feasible region, which is the set of all possible solutions that satisfy the constraints. The optimal solution will lie at one of the vertices (corner points) of the feasible region. This is a fundamental concept in linear programming.
Conclusion
In this article, we have demonstrated how to solve a system of linear equations using the elimination method and how to graphically represent the equations and the feasible region. We solved the equations 3x - 4y = 7 and x - 4y = 3, finding the solution to be x = 2 and y = -1/4. We also discussed how to graph these lines and how to determine the feasible region when dealing with inequalities.
Understanding these concepts is essential for solving various mathematical problems and for applying linear equations in real-world scenarios. Graphical representation provides a visual aid that enhances our understanding and makes problem-solving more intuitive.
Whether you're a student learning algebra or a professional working in a quantitative field, mastering the techniques discussed in this article will be a valuable asset.
By practicing these methods and exploring different systems of equations and inequalities, you can further develop your skills and gain a deeper understanding of linear algebra and its applications. Remember that the key to success in mathematics is consistent practice and a willingness to explore new concepts and techniques.
