Solving Problems By Elimination

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 …………… (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.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.

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The elimination method of solving systems of equations is also called the addition method.

To solve a system of equations by elimination we transform the system such that one variable "cancels out".

Multiplication can be used to set up matching terms in equations before they are combined.

When using the multiplication method, it is important to multiply all the terms on both sides of the equation—not just the one term you are trying to eliminate.

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.

You can use this Elimination Calculator to practice solving systems.

So if you have a system: x – 6 = −6 and x y = 8, you can add x y to the left side of the first equation and add 8 to the right side of the equation.

Example 3: $$ \begin 2x - 5y &= 11 \ 3x 2y &= 7 \end $$ Solution: In this example, we will multiply the first row by -3 and the second row by 2; then we will add down as before.

$$ \begin &2x - 5y = 11 \color\ &\underline \end\ \begin &\underline} \text\ &19y = -19 \end $$ Now we can find: back into first equation: $$ \begin 2x - 5\color &= 11 \ 2x - 5\cdot\color &= 11\ 2x 5 &= 11\ \color &\color \color \end $$ The solution is $(x, y) = (3, -1)$.

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