Solving five unknown simultaneous equation by using EXCEL

Solving simultaneous equation by using EXCEL Inverse Matrix calculation in EXCEL Matrix multiplication in EXCEL EXCEL Tips Easy equation solving https://www.cliffsnotes.com/study-guides/algebra/algebra-i/equations-with-two-variables/solving-systems-of-equations-simultaneous-equations Solving Systems of Equations (Simultaneous Equations) If you have two different equations with the same two unknowns in each, you can solve for both unknowns. There are three common methods for solving: addition/subtraction, substitution, and graphing. Addition/subtraction method This method is also known as the elimination method. To use the addition/subtraction method, do the following: Multiply one or both equations by some number(s) to make the number in front of one of the letters (unknowns) the same or exactly the opposite in each equation. Add or subtract the two equations to eliminate one letter. Solve for the remaining unknown. Solve for the other unknown by inserting the value of the unknown found in one of the original equations. Example 1 Solve for x and y. Adding the equations eliminates the y‐terms. Now inserting 5 for x in the first equation gives the following: Answer: x = 5, y = 2 By replacing each x with a 5 and each y with a 2 in the original equations, you can see that each equation will be made true. In Example and Example , a unique answer existed for x and y that made each sentence true at the same time. In some situations you do not get unique answers or you get no answers. You need to be aware of these when you use the addition/subtraction method. Example 2 Solve for x and y. First multiply the bottom equation by 3. Now the y is preceded by a 3 in each equation. The equations can be subtracted, eliminating the y terms. Insert x = 5 in one of the original equations to solve for y. Answer: x = 5, y = 3 Of course, if the number in front of a letter is already the same in each equation, you do not have to change either equation. Simply add or subtract. To check the solution, replace each x in each equation with 5 and replace each y in each equation with 3. Example 3 Solve for a and b. Multiply the top equation by 2. Notice what happens. Now if you were to subtract one equation from the other, the result is 0 = 0. This statement is always true. When this occurs, the system of equations does not have a unique solution. In fact, any a and b replacement that makes one of the equations true, also makes the other equation true. For example, if a = –6 and b = 5, then both equations are made true. [3(– 6) + 4(5) = 2 AND 6(– 6) + 8(5) = 4] What we have here is really only one equation written in two different ways. In this case, the second equation is actually the first equation multiplied by 2. The solution for this situation is either of the original equations or a simplified form of either equation. Example 4 Solve for x and y. Multiply the top equation by 2. Notice what happens. Now if you were to subtract the bottom equation from the top equation, the result is 0 = 1. This statement is never true. When this occurs, the system of equations has no solution. In Examples 1–4, only one equation was multiplied by a number to get the numbers in front of a letter to be the same or opposite. Sometimes each equation must be multiplied by different numbers to get the numbers in front of a letter to be the same or opposite. Solve for x and y. Notice that there is no simple number to multiply either equation with to get the numbers in front of x or y to become the same or opposites. In this case, do the following: Select a letter to eliminate. Use the two numbers to the left of this letter. Find the least common multiple of this value as the desired number to be in front of each letter. Determine what value each equation needs to be multiplied by to obtain this value and multiply the equation by that number. Suppose you want to eliminate x. The least common multiple of 3 and 5, the number in front of the x, is 15. The first equation must be multiplied by 5 in order to get 15 in front of x. The second equation must be multiplied by 3 in order to get 15 in front of x. Now subtract the second equation from the first equation to get the following: At this point, you can either replace y with and solve for x (method 1 that follows), or start with the original two equations and eliminate y in order to solve for x (method 2 that follows). Method 1 Using the top equation: Replace y with and solve for x. Method 2 Eliminate y and solve for x. The least common multiple of 4 and 6 is 12. Multiply the top equation by 3 and the bottom equation by 2. Now add the two equations to eliminate y. The solution is x = 1 and . Substitution method Sometimes a system is more easily solved by the substitution method. This method involves substituting one equation into another.