Solving Systems Of Equations 8x-3y=20 And 3x=21
When faced with a system of equations, the goal is to find the values of the variables that satisfy all equations simultaneously. In this article, we will delve into the process of solving the given system of equations:
8x - 3y = 20
3x = 21
We will explore the methods used to find the solution and, in case of a dependent system, express the solution set in terms of x. Understanding how to solve systems of equations is a fundamental skill in algebra, with applications in various fields, including engineering, economics, and computer science. By mastering these techniques, you can tackle complex problems involving multiple variables and relationships.
Method 1: Substitution
One of the most common methods for solving systems of equations is substitution. This method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation with one variable, which can then be easily solved. Once you find the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. This process allows you to systematically eliminate variables and find the solution that satisfies both equations simultaneously. The substitution method is particularly useful when one of the equations can be easily solved for one variable in terms of the other.
Step 1: Solve for x in the Second Equation
The given system of equations is:
8x - 3y = 20
3x = 21
The second equation, 3x = 21, is simpler and can be easily solved for x. To isolate x, we divide both sides of the equation by 3:
3x / 3 = 21 / 3
x = 7
Thus, we find that x equals 7. This is a crucial first step in solving the system, as it provides a concrete value for one of the variables. Knowing the value of x allows us to substitute it into the first equation and solve for y, effectively reducing the problem to a single equation with one unknown.
Step 2: Substitute x = 7 into the First Equation
Now that we have found the value of x, we can substitute it into the first equation to solve for y. The first equation is:
8x - 3y = 20
Substituting x = 7 into this equation, we get:
8(7) - 3y = 20
56 - 3y = 20
This substitution is a key step in the substitution method, as it allows us to eliminate one variable and focus on solving for the other. By replacing x with its numerical value, we transform the equation into one that involves only y, making it straightforward to isolate and solve for y.
Step 3: Solve for y
We now have the equation:
56 - 3y = 20
To isolate y, we first subtract 56 from both sides of the equation:
56 - 3y - 56 = 20 - 56
-3y = -36
Next, we divide both sides by -3:
-3y / -3 = -36 / -3
y = 12
Thus, we find that y equals 12. This step completes the process of solving for both variables in the system of equations. We now have values for both x and y, which can be verified by substituting them back into the original equations.
Step 4: Verify the Solution
To ensure our solution is correct, we substitute the values x = 7 and y = 12 back into the original equations:
For the first equation:
8x - 3y = 20
8(7) - 3(12) = 20
56 - 36 = 20
20 = 20
For the second equation:
3x = 21
3(7) = 21
21 = 21
Both equations hold true, so our solution is correct. Verification is a crucial step in solving systems of equations, as it helps to catch any errors made during the process and ensures that the solution satisfies all the given conditions. By substituting the values back into the original equations, we can confidently confirm the accuracy of our solution.
Method 2: Elimination
The elimination method is another powerful technique for solving systems of equations. This method involves manipulating the equations so that the coefficients of one of the variables are opposites (i.e., one is the negative of the other). When the equations are added together, the variable with opposite coefficients is eliminated, resulting in a single equation with one variable. This equation can then 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. The elimination method is particularly useful when the coefficients of one of the variables are already opposites or can be easily made opposites by multiplying one or both equations by a constant. This method provides a systematic way to eliminate variables and solve the system.
Step 1: Observe the Equations
The given system of equations is:
8x - 3y = 20
3x = 21
In this case, we can see that the second equation only involves x. This makes the elimination method less directly applicable, as we don't have a y term to eliminate. However, we can still use the value of x obtained from the second equation to solve for y in the first equation, effectively using a combination of elimination and substitution.
Step 2: Solve for x in the Second Equation
As we did in the substitution method, we solve the second equation for x:
3x = 21
x = 21 / 3
x = 7
This step is identical to the first step in the substitution method and provides us with the value of x. Knowing the value of x is crucial for proceeding with either the substitution or elimination method, as it allows us to reduce the system to a single equation with one unknown.
Step 3: Substitute x = 7 into the First Equation
We substitute x = 7 into the first equation:
8x - 3y = 20
8(7) - 3y = 20
56 - 3y = 20
This step is the same as in the substitution method and sets up the equation for solving for y. By replacing x with its numerical value, we eliminate one variable and create an equation that involves only y.
Step 4: Solve for y
We now solve for y:
56 - 3y = 20
-3y = 20 - 56
-3y = -36
y = -36 / -3
y = 12
Thus, we find that y equals 12. This step completes the process of solving for both variables in the system of equations, using a combination of solving for one variable and substituting its value into the other equation. This approach is a common adaptation of the elimination method when one equation is simpler and can be easily solved for one variable.
Step 5: Verify the Solution
To verify our solution, we substitute x = 7 and y = 12 back into the original equations:
For the first equation:
8x - 3y = 20
8(7) - 3(12) = 20
56 - 36 = 20
20 = 20
For the second equation:
3x = 21
3(7) = 21
21 = 21
Both equations hold true, confirming that our solution is correct. Verification is a crucial step in solving systems of equations, as it helps to catch any errors and ensures that the solution satisfies all the given conditions.
Solution
The solution to the system of equations is x = 7 and y = 12. This means that the point (7, 12) is the intersection of the two lines represented by the equations. The solution is unique, indicating that the system is independent and consistent.
Conclusion
In this article, we successfully solved the system of equations:
8x - 3y = 20
3x = 21
We used both the substitution and elimination methods to arrive at the solution x = 7 and y = 12. Understanding these methods is crucial for solving various mathematical problems and real-world applications. Mastering these techniques allows you to tackle complex systems of equations with confidence and accuracy. Whether you choose substitution or elimination, the key is to systematically reduce the system to a single equation with one variable, solve for that variable, and then substitute back to find the other variable. By practicing these methods, you can develop a strong foundation in algebra and problem-solving.
