Question

There's a question I've been trying for 2 days to get solved but either I'm missing something or maybe I haven't been using the right method but I hope someone will help me here. ________________________________________

Substitution, Manipulation, Elimination

Solve Simultaneously. Use the method you find easiest. (title of the question)

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4x-3y=11

5x-9y=-2

Answer 1

(x, y) = (5, 3)

Explanation:

You want the solution to the system of equations ...

• 4x -3y = 11
• 5x -9y = -2

Solution

It is convenient to subtract the second equation from 3 times the first:

3(4x -3y) -(5x -9y) = 3(11) -(-2)

12x -9y -5x +9y = 33 +2

7x = 35 . . . . . . . simplify

x = 5 . . . . . . . . . . divide by 7

4(5) -3y = 11 . . . . . substitute into the first equation

9 = 3y . . . . . . . . add 3y-11 to both sides

y = 3 . . . . . . . . divide by 3

The solution is (x, y) = (5, 3).

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Additional comment

We find using a graphing calculator to be the easiest way to solve a pair of simultaneous equations. The attachment shows the solution is the one we found above.

The approach of "substitution" is straightforward, if error-prone. Basically, you solve one equation for either variable, then use that expression in the other equation. Here, for example, you can solve the first for x:

x = (11 +3y)/4

Then use that in the second equation:

5(11 +3y)/4 -9y = -2

5(11 +3y) -36y = -8 . . . . eliminate the denominator

55 +15y -36y = -8 . . . . . eliminate the parentheses

-21y = -63 . . . . . . . . . . . simplify, subtract 55

y = 3 . . . . . . . . . . . . divide by -21

x = (11 +3(3))/4 = 20/4 = 5 . . . . find x

In the above, we used "elimination." We took advantage of the fact that the y-coefficients were related by a factor of 3. To cancel y-terms, we need to have the equations we "add" have opposite signs for the y-terms. Here, we do that by multiplying the first by 3 (to make -9y), then subtracting the second equation (which has -9y, so will cancel). We could have subtracted 3 times the first from the second, but that would make all the resulting coefficients be negative, which we like to avoid.

Or, we could have multiplied the first equation by -3 to make the y-coefficients opposite, then added the results. (Again, that would give negative coefficients in the sum.) Planning ahead can help avoid mistakes due to minus signs.