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If you don't have equations where you can eliminate a variable by addition or subtraction you directly you can begin by multiplying one or both of the equations with a constant to obtain an equivalent linear system where you can eliminate one of the variables by addition or subtraction.Use elimination when you are solving a system of equations and you can quickly eliminate one variable by adding or subtracting your equations together.
One child ticket costs $4.50 and one adult ticket costs $6.00. Adding or subtracting two equations in order to eliminate a common variable is called the elimination (or addition) method.
Once one variable is eliminated, it becomes much easier to solve for the other one.
Example 2: $$ \begin x 3y &= -5 \\ 4x - y &= 6 \end $$ Solution: Look at the x - coefficients.
Multiply the first equation by -4, to set up the x-coefficients to cancel.
Substituting the value of y = 3 in equation (i), we get 2x 3y = 11 or, 2x 3 × 3 = 11or, 2x 9 = 11 or, 2x 9 – 9 = 11 – 9or, 2x = 11 – 9or, 2x = 2 or, x = 2/2 or, x = 1Therefore, x = 1 and y = 3 is the solution of the system of the given equations. Solve 2a – 3/b = 12 and 5a – 7/b = 1 Solution: The given equations are: 2a – 3/b = 12 ……………
Solving Problems By Elimination Effective Business Plans
(iv) Multiply equation (iii) by 5 and (iv) by 2, we get 10a – 15c = 60 …………… (vi) Subtracting (v) and (vi), we get or, c = 58 /-29 or, c = -2 But 1/b = c Therefore, 1/b = -2 or b = -1/2 Subtracting the value of c in equation (v), we get 10a – 15 × (-2) = 60 or, 10a 30 = 60 or, 10a 30 - 30= 60 - 30 or, 10a = 60 – 30 or, a = 30/10 or, a = 3 Therefore, a = 3 and b = 1/2 is the solution of the given system of equations. x/2 2/3 y = -1 and x – 1/3 y = 3 Solution: The given equations are: x/2 2/3 y = -1 …………… (ii) Multiply equation (i) by 6 and (ii) by 3, we get; 3x 4y = -6 …………… (iv) Solving (iii) and (iv), we get; or, y = -15/5 or, y = -3 Subtracting the value of y in (ii), we get; x - 1/3̶ × -3̶ = 3 or, x 1 = 3 or, x = 3 – 1 or, x = 2 Therefore, x = 2 and y = -3 is the solution of the equation.This method for solving a pair of simultaneous linear equations reduces one equation to one that has only a single variable.Once this has been done, the solution is the same as that for when one line was vertical or parallel.And since x y = 8, you are adding the same value to each side of the first equation.If you add the equations above, or add the opposite of one of the equations, you will get an equation that still has two variables.Recall that a false statement means that there is no solution.If both variables are eliminated and you are left with a true statement, this indicates that there are an infinite number of ordered pairs that satisfy both of the equations. A theater sold 800 tickets for Friday night’s performance. Combining equations is a powerful tool for solving a system of equations.When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.Example $$\begin 3y 2x=6\ 5y-2x=10 \end$$ We can eliminate the x-variable by addition of the two equations.So let’s now use the multiplication property of equality first.You can multiply both sides of one of the equations by a number that will result in the coefficient of one of the variables being the opposite of the same variable in the other equation. Notice that the first equation contains the term 4y, and the second equation contains the term y.