How do we solve the solution to the following by elimination?  3x-2y=11, 6x+11y=97thank you

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mlsiasebs | College Teacher | (Level 1) Associate Educator

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To solve a linear system by the elimination method, we need to add the two equations together in a way such that one of the variables is eliminated from the answer.

For this problem, neither the coefficient for x nor y have the same value.  When looking at the equations, we can look at either x or y to focus on eliminating that variable.  For this problem, I see that the coefficients of x are 3 and 6 making it easy to find a common value so we'll focus on the x.

I need the coefficients of x to have the same value but a different sign which means I'll need to multiply the first equation by -2.  Remember in algebra whatever we do to one side of an equation we must also do to the other.

So, 3x - 2y = 11 becomes

-6x + 4y = -22

since we need to multiply each number by -2.  Now we can take our new equation and add it to the second equation.

-6x + 4y = -22

6x + 11y = 97


15 y = 75

The x terms were eliminated so we are left with 15y = 75 which we can solve to find

y = 5

Now that we know the value of y, we can plug it back into one of our original equations (you can use either one) to solve for the value of x

3x - 2y = 11

3x - 2(5) = 11

3x -10 = 11

3x = 21

x = 7

As a way of checking ourself, we can substitute both values into the second equation and see if it is true.

6x + 11y = 97

6(7) + 11(5) = 97

42 + 55 = 97

97 = 97

so x = 7 and y = 5 are the correct solution to this system.