Substitute this value in any of the two equations given and find the value of the other unknown. Solve the following system of equations by the Method of Elimination Example: Solve for and Solution: … 1 … 2 Step 1: We multiply equation 1 by 5 and equation 2 by 3. The resulting equations are: … 3 … 4 Step 2: Subtracting equation 4 from equation 3 we have, or, Step3: or, Step 4: Substituting in equation 1 we get, or, or, the required solution is: Note: Here we have chosen the numbers to be multiplied such that the coefficients of become equal and we can thus eliminate and arrive at an equation only in as can be seen in steps 2 and 3.
This is shown below: … 1 … 2 We multiply equation 1 by 3 and equation 2 by 4. The resulting equations are: … 3 … 4 Subtracting equation 4 from equation 3 we have, or, or, Substituting in equation 1 we have, or, or, Hence, it can be safely concluded that, which variable we choose to eliminate will not alter the solution in any way.
Example: Solve for and Solution: … 1 … 2 Step 1: We multiply equation 2 by 2. The resulting equations are: … 3 … 1 Step 2: Adding equations 3 and 1 we have, or, Step 3: Step 4: Substituting this value of in equation 1 we have, or, or, the required solution is: Note: It should be kept in mind that whatever method is undertaken for solving the pair of simultaneous equations the solution will remain the same, i.
Method of Comparison Steps: From the first equation find the value of any one unknown quantity in terms of the other. From the second equation find the value of the same unknown quantity in terms of the other. For example, say we have a pair of simultaneous equations in and. From the first equation we express the value of in terms of. From the second equation also, we express the value of in terms of. These two values so obtained are obviously equal because and for all values of and.
Equate these two values to obtain an equation in one variable and solve for that variable. Substituting this value of one of the variables in any one of the earlier equations, find out the value of the other variable. Such equations are solved by the method, given below: … 1 … 2 On adding 1 and 2 , we get: or, … 3 dividing each term by 98 On subtracting 2 from 1 , we get: Or, … 4 dividing each term by 32 Now on solving equations 3 and 4 , we get: Problems leading to Simultaneous Equations To solve a problem on simultaneous equations, adopt the following steps: Assume the two variables unknowns as and.
According to the problem, set up two equations in terms of and. Solve the pair of simultaneous equations by any of the methods that have been explained in this article and the other article on simultaneous equations.
On reversing the digits, the number becomes According to the problem: On solving we get: [Here any one of the methods illustrated may be used to solve these equations] the required number Example: A and B each have certain number of oranges. Solution: Let A has number of oranges and B has number of oranges.
In 1 st case if B gives oranges to A : or, … 1 In 2 nd case If A gives 10 oranges to B or, … 2 On solving equations 1 and 2 , we get: and. Exercise Solve the following pairs of linear simultaneous equations by the method of comparison: The sides of an equilateral triangle are given by and respectively.
Find the lengths of the sides of the triangle. A and B both have pencils. If A gives 10 pencils to B, then B will have twice as many as A and if B gives 10 pencils to A, then they will have the same number of pencils.
How many pencils does each have? In an examination the ratio of passes to failures was Had 30 less appeared and 20 less passed, the ratio of passes to failures would have been Find the number of students who appeared for the examination. A farmer wishing to purchase a certain number of sheep found that if they cost him Rs 42 a head, he would not have money enough by Rs 28, but if they cost him Rs 40 a head, he would then have Rs 40 more than what he required.
Find the number of sheep and the money which he had? You want to find out the better deal when renting a car, and you're comparing two rental companies. By putting the variable and fixed costs, such as the per-mile and daily rate, into an algebraic expression, then solving for the total cost, you can see which company saves you money for different amounts of driving. You can use this same process with a system of equations when trying to decide on the best cell phone plan, determining at how many minutes both companies charge the same amount and deciding from there which is the best plan for you and your intended usage.
Simultaneous equations can be used to determine the best loan choice to make when buying a car or a house when you consider the duration of the loan, the interest rate and the monthly payment of the loan. Other variables may be involved as well. With the information at hand, you can calculate which loan is the best choice for you.
Simultaneous equations can be used when considering the relationship between the price of a commodity and the quantities of the commodity people want to buy at a certain price. An equation can be written that describes the relationship between quantity, price and other variables, such as income. These relationship equations can be solved simultaneously to determine the best way to price the commodity and sell it. To find the point of intersection, you multiply one of the equations by a number.
This number is chosen so that one of the following is true:. For case A , we subtract one equation from the other, term by term:. Both cases produce the same solution.
For this particular example, finding the point of intersection graphically is not too difficult. However, what if we wish to solve.
As you can see, finding the intersection point graphically is not as straightforward as above and could result in a lack of precision. This demonstrates how it is important to know how to solve paired linear equations using the above method, rather than using graphs.
For most cases this will be true.
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