Solve The System Of Equations A Step-by-Step Guide

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In the realm of mathematics, solving systems of equations is a fundamental skill with applications spanning various fields, from engineering and physics to economics and computer science. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. In simpler terms, we are looking for the point(s) where the lines or curves represented by the equations intersect.

Why are systems of equations important? They allow us to model and solve real-world problems involving multiple variables and constraints. For example, we might use a system of equations to determine the equilibrium price and quantity in a market, to design a bridge that can withstand certain loads, or to optimize the allocation of resources in a business. Mastering the techniques for solving systems of equations is therefore crucial for anyone pursuing a career in a STEM field or any discipline that relies on quantitative analysis.

Methods for Solving Systems of Equations

There are several methods for solving systems of equations, each with its advantages and disadvantages. The most common methods include:

  • Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation in one variable, which can then be solved. The value of that variable is then substituted back into one of the original equations to find the value of the other variable.
  • Elimination (or Addition/Subtraction): This method involves manipulating the equations so that the coefficients of one of the variables are opposites. Adding the equations then eliminates that variable, leaving a single equation in one variable. This equation can be solved, and the value of the variable can be substituted back into one of the original equations to find the value of the other variable.
  • Graphing: This method involves graphing the equations on the same coordinate plane. The solutions to the system are the points where the graphs intersect. This method is particularly useful for visualizing the solutions and understanding the nature of the system.
  • Matrices: For systems with more than two variables, matrix methods such as Gaussian elimination and matrix inversion can be more efficient. These methods involve representing the system of equations in matrix form and using matrix operations to solve for the variables.

The choice of method depends on the specific system of equations and personal preference. For simple systems with two variables, substitution and elimination are often the most efficient methods. For more complex systems, matrix methods may be necessary.

1. Solving Systems of Equations by Substitution

The substitution method is a powerful technique for solving systems of equations. The core idea behind this method is to express one variable in terms of the other, effectively reducing the system to a single equation with a single unknown. This makes the equation solvable, and once we find the value of one variable, we can easily substitute it back into one of the original equations to find the value of the other. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other.

The steps involved in solving a system of equations using the substitution method are as follows:

  1. Solve one equation for one variable: Choose the equation that is easiest to solve for one of the variables. This usually means picking the equation where a variable has a coefficient of 1 or -1. Isolate that variable on one side of the equation. For instance, if you have the equation x + 2y = 5, it's straightforward to solve for x as x = 5 - 2y.
  2. Substitute the expression into the other equation: Take the expression you found in step 1 and substitute it into the other equation. This will result in an equation that contains only one variable. For example, if your other equation is 3x - y = 2, substitute (5 - 2y) for x to get 3(5 - 2y) - y = 2.
  3. Solve the new equation: Solve the equation you obtained in step 2 for the remaining variable. This may involve simplifying, combining like terms, and isolating the variable. In our example, we would solve 3(5 - 2y) - y = 2 for y.
  4. Substitute back to find the other variable: Once you have the value of one variable, substitute it back into either of the original equations or the expression you found in step 1 to solve for the other variable. Choose the equation that looks easiest to work with. If we found y = 1 in the previous step, we could substitute it back into x = 5 - 2y to find x = 5 - 2(1) = 3.
  5. Check your solution: Always check your solution by substituting the values you found for both variables into both original equations. If the solution satisfies both equations, it is correct. This step is crucial to avoid errors. In our example, we would check if x = 3 and y = 1 satisfy both x + 2y = 5 and 3x - y = 2.

Let's illustrate this with an example. Consider the system of equations:

x + y = 10
2x - y = 5

We can easily solve the first equation for x: x = 10 - y. Now, substitute this expression for x into the second equation:

2(10 - y) - y = 5
20 - 2y - y = 5
20 - 3y = 5
-3y = -15
y = 5

Now that we have y = 5, substitute it back into the expression for x: x = 10 - 5 = 5. Therefore, the solution to the system is x = 5 and y = 5. We can check this by plugging these values into the original equations:

5 + 5 = 10 (True)
2(5) - 5 = 5 (True)

The substitution method can be particularly useful when dealing with nonlinear systems of equations or systems where one equation is already solved for one variable.

2. Solving Systems of Equations by Elimination

The elimination method, also known as the addition or subtraction method, provides an alternative approach to solving systems of equations. This technique focuses on manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. This leads to a simpler equation with only one variable, which can then be solved. The value of the eliminated variable can then be found by substituting the value of the solved variable back into one of the original equations. The elimination method shines when the coefficients of one variable in the two equations are the same or easily made the same by multiplication.

The steps involved in solving a system of equations using the elimination method are as follows:

  1. Align the equations: Write the equations one above the other, aligning the variables (x terms above x terms, y terms above y terms) and the constant terms. This helps to visually organize the equations and makes the next steps easier.
  2. Multiply one or both equations (if necessary): Look at the coefficients of the variables. If necessary, multiply one or both equations by a constant so that the coefficients of either x or y are opposites (same absolute value, different signs) or the same. The goal is to have coefficients that will cancel out when the equations are added or subtracted. For example, if you have 2x + 3y = 7 and x - y = 1, you could multiply the second equation by -2 to get -2x + 2y = -2. Now the x coefficients are opposites.
  3. Add or subtract the equations: Add the equations together if the coefficients of one variable are opposites. Subtract the equations if the coefficients of one variable are the same. This will eliminate one of the variables, leaving you with a single equation in one variable. In our example, adding 2x + 3y = 7 and -2x + 2y = -2 gives 5y = 5.
  4. Solve the new equation: Solve the resulting equation for the remaining variable. This is a straightforward algebraic step. From 5y = 5, we get y = 1.
  5. Substitute back to find the other variable: Substitute the value you found in step 4 back into either of the original equations to solve for the other variable. Choose the equation that looks easier to work with. If we substitute y = 1 into x - y = 1, we get x - 1 = 1, so x = 2.
  6. Check your solution: As with the substitution method, it is crucial to check your solution by substituting the values you found for both variables into both original equations. If the solution satisfies both equations, it is correct. Checking our solution x = 2 and y = 1 in the original equations 2x + 3y = 7 and x - y = 1 confirms that it is correct.

Consider another example:

3x + 2y = 16
7x - 2y = 4

Notice that the coefficients of y are already opposites. Therefore, we can add the equations directly:

(3x + 2y) + (7x - 2y) = 16 + 4
10x = 20
x = 2

Now, substitute x = 2 into the first equation:

3(2) + 2y = 16
6 + 2y = 16
2y = 10
y = 5

Thus, the solution is x = 2 and y = 5. This method is particularly efficient when the equations are already set up nicely for elimination, or when the coefficients can be easily manipulated to facilitate elimination.

3. Solving Systems of Equations by Graphing

The graphing method provides a visual approach to solving systems of equations. This method involves plotting the graphs of each equation in the system on the same coordinate plane. The solution(s) to the system correspond to the point(s) where the graphs intersect. This is because the points of intersection represent the ordered pairs (x, y) that satisfy both equations simultaneously. The graphing method is particularly useful for visualizing the solutions and understanding the nature of the system – whether it has one solution, infinitely many solutions, or no solution.

The steps involved in solving a system of equations using the graphing method are as follows:

  1. Rewrite the equations in slope-intercept form (y = mx + b): This form makes it easy to identify the slope (m) and y-intercept (b) of each line, which are needed for graphing. If the equations are not linear, you'll need to find other ways to plot their graphs, such as by plotting points or using a graphing calculator.
  2. Graph each equation: Plot each line on the same coordinate plane. Use the slope and y-intercept to plot the lines, or create a table of values and plot the points. For nonlinear equations, you may need to plot more points to get an accurate representation of the graph.
  3. Identify the point(s) of intersection: The points where the graphs intersect represent the solutions to the system of equations. Read the coordinates (x, y) of these points from the graph. These coordinates are the values of x and y that satisfy both equations.
  4. Check your solution(s): Substitute the coordinates of the intersection point(s) into both original equations to verify that they satisfy both equations. This is an essential step to ensure that the solution(s) you found graphically are accurate.

Let's consider an example. Solve the following system of equations graphically:

y = x + 1
y = -x + 3

Both equations are already in slope-intercept form. The first equation has a slope of 1 and a y-intercept of 1. The second equation has a slope of -1 and a y-intercept of 3. Plotting these lines on a coordinate plane, we find that they intersect at the point (1, 2). Thus, the solution to the system is x = 1 and y = 2. We can check this by substituting these values into the original equations:

2 = 1 + 1 (True)
2 = -1 + 3 (True)

Now, let's consider a system with no solution:

y = 2x + 1
y = 2x - 3

These lines have the same slope (2) but different y-intercepts, meaning they are parallel and will never intersect. Therefore, this system has no solution. Graphically, you would see two parallel lines.

And, consider a system with infinitely many solutions:

x + y = 3
2x + 2y = 6

If we divide the second equation by 2, we get x + y = 3, which is the same as the first equation. This means the two equations represent the same line. Graphically, they overlap completely, and every point on the line is a solution. This system has infinitely many solutions.

While the graphing method provides a clear visual representation of the solutions, it may not be the most accurate method for systems with solutions that are not integers. In such cases, algebraic methods like substitution or elimination may provide more precise solutions.

Applying the Methods to a Specific System

Now, let's apply these methods to solve the given system of equations:

-9y + 4x - 11 = 0
-3y + 10x + 31 = 0

1. Rearrange the Equations

First, it's helpful to rearrange the equations into a more standard form (Ax + By = C):

4x - 9y = 11  (Equation 1)
10x - 3y = -31 (Equation 2)

2. Solving Using the Elimination Method

To use the elimination method, we need to make the coefficients of either x or y opposites or the same. Let's eliminate y. We can multiply Equation 1 by -1 and Equation 2 by 3:

-1 * (4x - 9y) = -1 * 11  =>  -4x + 9y = -11
3 * (10x - 3y) = 3 * -31 =>  30x - 9y = -93

Now, add the two equations:

(-4x + 9y) + (30x - 9y) = -11 + (-93)
26x = -104
x = -4

3. Substitute to Find y

Substitute x = -4 into either Equation 1 or Equation 2. Let's use Equation 1:

4(-4) - 9y = 11
-16 - 9y = 11
-9y = 27
y = -3

4. Check the Solution

Check the solution x = -4 and y = -3 in both original equations:

Equation 1: -9(-3) + 4(-4) - 11 = 27 - 16 - 11 = 0 (True) Equation 2: -3(-3) + 10(-4) + 31 = 9 - 40 + 31 = 0 (True)

Therefore, the solution is x = -4 and y = -3.

5. Provide the Answer

x = -4
y = -3

Conclusion

Solving systems of equations is a crucial skill in mathematics with broad applications. The substitution, elimination, and graphing methods offer different approaches to finding solutions. Each method has its strengths and is suitable for different types of systems. By mastering these techniques, you can tackle a wide range of problems involving multiple variables and constraints. In this article, we explored each method in detail and applied them to a specific system, demonstrating the step-by-step process of finding the solution.

Solve the following system of equations:

-9y + 4x - 11 = 0
-3y + 10x + 31 = 0

Find the values of x and y that satisfy both equations.

Solving Systems of Equations Step by Step Guide with Examples