Find the x, y (Two different equations) Thank you!!?

Question

-x - y = 7
-3x + 5y = -11


-5x + y = 11
-4x + 5y = 13

Answer 1

Hi,

You already have an answer showing you how to solve these by substitution, but the first problem lost a negative sign. It should have been:

To solve by substitution, solve an equation for one of its variables. It's generally easier to solve if you pick a variable with a 1 or -1 as its coefficient. In this case, let's take the first equation and solve it for x.

-x - y = 7 Add y.
-x = y + 7 Divide by -1
x = -y - 7 This is the expression you will now substitute in the OTHER equation. (If you substitute it back in the equation you already used, it will make the variables both disappear. It's hard to solve for them that way.)

-3x + 5y = -11 Substituting -y - 7 for x, you get:
-3(-7-y) + 5y = -11
21 + 3y + 5y =-11
8y +21 = -11
8y = -32
y = -4

Now substitute this back into either equation and solve for x.

-3x + 5y = -11
-3x + 5(-4) = -11
-3x - 20 = -11
-3x = 9
x = -3

So the answer for this is (-3,-4).

Another method to solve the same problem is to use addition/subtraction. It's also called linear combination. To do that, pick which letter you want to eliminate. Either letter will work. For the first problem, let's eliminate the "y". Look at the coefficients ox "y". You want to find a number that they both can divide into and then multiply one or both equations so that those coefficients become plus and minus that number. For our equations
-x - y = 7
-3x + 5y = -11
1 and 5 from in front of y both go into 5, so you want one equation to have + 5y and the other to have a -5y. The second equation already has a + 5y, so it's fine the way it is. To get a -5y in the first equation, the entire equation needs multiplied by 5.

5(-x - y = 7)
-3x + 5y = -11

Now your equations are:

-5x - 5y = 35
-3x + 5y = -11
-------------------- Now add these equations together.
-8x = 24 Divide by -8
x = -3

Just like we did before, substitute this back into either equation and solve for y.

-3x + 5y = -11
-3(-3) + 5y = -11
9 + 5y = -11
5y = -20
y = -4

We get the same answer as the other method gave us, the point (-3,-4).

For the second problem, done first by substitution, let's solve the top equation for y. That's the only variable with a coefficient of 1, so that makes the problem easier to solve without dealing with fractions.

-5x + y = 11 Add 5x.
y = 5x + 11 This is the expression to substitute into the OTHER equation, that was not used yet.

-4x + 5y = 13 Substitute 5x + 11 in place of y:
-4x + 5(5x + 11) = 13
-4x + 25x +55 = 13
21x + 55 = 13
21x = -42
x = -2

Just like we did before, substitute this back into either equation and solve for y.

-5x + y = 11 Let x = -2
-5(-2) + y = 11
10 + y = 11
y = 1

The answer to your second equation is (-2,1)

To solve this equation by addition/subtraction or linear combination, let's choose this time to eliminate the "x" terms. Because they have -5 and -4 as coefficients, those both can divide into 20. That means you want to get 20x in one equation and -20x in the other by multiplying your equations like this:

-4(-5x + y = 11)
5(-4x + 5y = 13)

20x - 4y = -44
-20x + 25y = 65 Now add these together.
-----------------------
21y = 21
y = 1

Now substitute this back into either equation and solve for x.

-5x + y = 11
-5x + 1 = 11
-5x = 10
x = -2
The answer to your second equation is (-2,1) again.

There are other methods that solve these too. You can also use graphing, determinants, matrices, or matrices on your calculator to solve these.

I hope this helps!

Answer 2

You can solve these types of problems in a few different ways, but my favorite method is called elimination.

For the first problem, multiply everything in the top equation by 5. Then add the result to the second equation. The result will no longer have a y in it.

-5x - 5y = 35
-3x + 5y = -11

Their sum is -8x = 24. So x = -3. To get y, just substitute -3 for x in one of the original equations.
-(-3) - y = 7
3 - y = 7
y = -4

So the solution is (-3, -4)

For the second problem, multiply the first equation by -5.

25x - 5y = -55
-4x + 5y = 13

Their sum is 21x = -42 So x = -2.

-5(-2) + y = 11
10 + y = 11
So y = 1

Solution: (-2, 1)

The reason why I prefer the elimination method as opposed to others is because you typically avoid fractions, which most people aren't very good at computing with.

Answer 3

elimination by addition method

-x - y = 7- - - - - - - - -Equation 1
- 3x + 5y = - 11- - - -Equation 2
- - - - - - - - -

Multiply equation 1 by 5

- x - y = 7

- 5(x) - 5(y) = 5(7)

- 5x - 5y = 35
-- - - - - - - - - - -

Elimination of y

- 5x - 5y = 35
- 3x + 5y = - 11
- - - - - - - - - - -

- 8x = 24

- 8x / - 8 = 24 / - 8

x = - 24/8

x = - 3

Insert the x value into equation 1

- (- 3) - y = 7

3 - y = 7

3 - y - 3 = 7 - 3

- y = 4

- 1(- y) = - 1(4)

y = - 4

Insert the y value into equation 1

- - - - - - - - - - - - - - - - - - - - - - - - -

- x - y = 7

-( - 3) - ( - 4 = 7

3 + 4 = 7

7 = 7

- - - - - - - -

Check for equation 2

- 3x + 5y = - 11

- 3(-3) + 5(- 4) = - 11

- (- 9) + (- 20) = - 11

9 - 20 = - 11

- 11 = - 11

- - - - - - - - - - -

The two equations balance

The sulotion set is { - 3, - 4 }

- - - - - - - - - -s-

Answer 4

Question 1
-5x - 5y = 35
-3x + 5y = - 11--------ADD
-8x = 24
x = - 3
9 + 5y = -11
5y = - 20
y = - 4
Solution is x = - 3, y = - 4

Question 2
25x - 5y = - 55
-4x + 5y = 13--------ADD
21x = - 42
x = - 2
10 + y = 11
y = 1
Solution is x = -2, y = 1

Answer 5

-x = 7+y
x= -7-y

-3(-7-y) + 5y = -11
21+3y+5y =11
8y = -10

y = -5/4 = -1.25
x= -7+1.25
x= -5.75

Answer:
y= -1.25
x= -5.75

-5x + y = 11
y = 11 + 5x
-4x + 5(11+5x) = 13
-4x + 55 + 25x = 13
21x = -42
x = -2
y = 11 + 5(-2)
y = 11-10
y = 1

Answer:
x = -2
y = 1