elimination....

Question

Justify each step in the solution path to solve the system of equations 

{4/5x+6y=15

{-x+18y=11

{4x+30y=75

{-x+18y=11

{4x+30y=75

{-4x+72=44

102y=119

y=7/6

4x+30(7/6)=75

{x=10

{y=7/6

Answer choices Given systems of equation, add the equation together, solve for y, solve for x, multiply an equation by 5, substitute the value of y into an equation, multiply an equation by 4

Answer 1

Before tackling this problem, it is helpful to review why we can manipulate both sides of an equation and still obtain the equality of both sides.

If I said, for example that:

4 + 5 = 9

If I multiply everything in the equation by 2 I would obtain:

8 + 10 = 18

This is still true of course, and we can always manipulate one side of the equation as long as we follow the same rules for the other side.

So, in the example given, in the first step we have:

{4/5x+6y=15 {4x+30y=75

------------>

{-x+18y=11 {-x+18y=11

Here, we are multiplying every term in the top equation by 5. Explicitly, we are multiplying (4/5)x on the left by 5 to yield 4x on the right. We are multiplying 6y on the left by 5 to yield 30y on the right. And so, we must also multiply our sum on the left (15) by 5 to yield 75 on the right.

Now, in the second step, we have:

{4x+30y=75 {4x+30y=75

------------>

{-x+18y=11 {-4x+72y=44

Here, we see that the top equation remains the same, but the bottom equation is different. We see that in the bottom equation we are multiplying every term by 4. We multiply the first term, -x by 4 to yield -4x on the right. We then multiply 18y by 4 to yield 72y (there is a typo in the question here, it says 72 but should say 72y.) Then, finally we multiply the sum, 11 by 4 to yield 44. You might ask why we came to these decisions so far, and the answer is because we need to isolate one of the variables first. So we are trying to balance these equations so that if we add them together, we will cancel out one of the variables completely. In the next step, we see that we can do exactly that.

Now, we can add the top and bottom equation together to eliminate the x-terms and isolate the y-terms. We will do this with each pair of terms at a time.

First, we have -4x + 4x = 0. Now you see why we did the above manipulations, to arrive at two x terms that can be easily cancelled out by adding these equations and we can then solve for y in the next step!

Then, we have 30y + 72y = 102y.

And finally, we have the sum of the sums, 44 + 75 which equals 119.

So all in all, we end up with:

102y = 119

The next step is simply solving for y. If 102 * y = 119 then we can divide both sides of the equation by 102 to get y by itself:

102y / 102 = 119 / 102 and this simplifies to: y = 7/6

Now that we have solved for y, we can use our solution for y in any of our equations we have with x in it.

Since y = 7/6, we can return to the equation we were dealing with before:

4x + 30y = 75 And we can now plug y = 7/6 into this equation!

4x + 30(7/6) = 75

Now we must isolate x.

First let's get the x term on one side of the equation and everything else on the other side. To achieve this, we can subtract 30(7/6) from both sides of the equation and that would yield:

4x = 75 - 30(7/6)

Now, we can use a calculator to determine what 30 * 7/6 is but we can also think about it in our heads. If

we divide 30 by 6, we get 5.

7/6 * 30 = 6/6 * 30 + 1/6 * 30

We know that 6/6 * 30 is just 30, and 1/6 of 30 is 5.

So, 7/6 * 30 = 35

Now, we can solve for x.

Going back to our equation, we have:

4x = 75 - 35

Which we can simplify to:

4x = 40

By dividing both sides by 4, we can isolate x and yield:

x = 40/4 = 10

Therefore, we have first solved for y and found it to equal 7/6

And then we have used the solution for y to solve for x and found that x was equal to 10.

I hope this helped :)

Answer 2

elimination....

Justify each step in the solution path to solve the system of equations 

{4/5x+6y=15 Given systems of equations

{-x+18y=11

{4x+30y=75 multiply first equation by 5

{-x+18y=11

{4x+30y=75

{-4x+72=44 multiply second equation by 4

102y=119 add the equations together

y=7/6 solve for y

4x+30(7/6)=75 substitute the value of y into third equation

{x=10 solve for x

{y=7/6