Solve the following systems using the substitution method. y= -x - 1, 5x - 4y= -32 and the second one would be x-6y= -2, 4x+y=7

Question

how would i solve these problems? use step by steps please

Answer 1

These are called simultaneous equations. In other words if you solve for x and y the same solution works for both equations. The way to solve them is to

1. put a variable by itself on one side of an equation (e.g., on the left side of the equal sign)
2. then substitute what's on the other side of the equation (the right side of the equation) in place of the variable in the other equation. It's easiest if you put the expression in parentheses.

Specifically:

y= -x - 1,

5x - 4y= -32

substitute (-x -1) for y in the second equation. You get:

5x - 4(-x -1)= -32

5x + 4x + 4 = -32

             9x  = -32 -4     (subtract 4  from both sides)

               9x = -36        (divide both sides by 9)

                 x = -4

Now that you know the value of x, you can calculate the value of y by substituting -4 for x. 

y= -x - 1

y= -(-4) - 1

y= +3

Now that you know y=3 and x=-4 you can check your work by substituting the values for the variables:

y   = -x - 1

+3 = -(-4)-1

+3 = +3

      5x - 4y= -32

5(-4) - 4(3)= -32

    -20 - 16 = -32

           -32 = -32

 

 

Answer 2

Hi Deanna;

y= -x - 1

5x - 4y= -32

Let's begin with the second equation...

5x - 4y= -32

Now, let's take note of the fact that the y is isolated in the first equation.  Henceforth, we can execute substitution...

5x - 4(-x-1)= -32

We have removed the y and replaced it with (-x-1).

Let's distribute...

5x-(-4x)-(-4)=-32

Subtracting a negative number is the same as adding a positive number.

5x+4x+4=-32

Let's combine like terms...

9x+4=-32

Let's continue to combine like terms by subtracting 4 from both sides...

-4+9x-4=-32-4

9x=-36

Let's divide both sides by 9...

(9x)/9=-36/9

x=-4

To find y...

y= -x - 1

y=-(-4)-1

y=4-1

y=3

Let's double-check...

5x - 4y= -32

5(-4)-4y=-32

-20-4y=-32

-4y=-12

y=-12/-4

A negative divided by a negative is positive...

y=12/4

y=6/2

y=3

 

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

x-6y= -2

4x+y=7

In neither equation is a variable isolated.  Let's isolate.

Let's begin with the first equation...

x-6y=-2

Let's add 6y to both sides...

x=6y-2

Now, the second equation...

4x+y=7

4(6y-2)+y=7

24y-8+y=7

25y-8=7

25y=15

y=15/25=3/5

Now, let find the value of x...

x-6y= -2

x-6(3/5)=-2

x-(18/5)=-2

x=-2+(18/5)

x=(-10/5)+(18/5)

x=8/5

Let's double-check...

4x+y=7

4x+3/5=7

4x=7-(3/5)

4x=(35/5)-(3/5)

4x=32/5

x=(32/5)(1/4)

x=32/20

x=16/10

x=8/5

Answer 3

For the first question:

y=-x-1

5x-4y=-32

 

First, solve one of the equations for either x or y (it doesn't matter--you can pick).

 

Since the top equation is already solved for y, I will leave it in that form in this example solution.

 

Next, substitute the expression for the variable you chose into the other equation (the one you haven't yet manipulated).

 

In my example solution, this means:

5x-4y=-32 becomes

5x-4(-x-1)=-32

 

Solve for (in this case) x:

 

5x+4x+4=-32

9x=-36

x=-4

 

Now, plug this numerical value for x into the expression for y:

y=-x-1

y=-(-4)-1

y=3

 

Your solution to the first problem is:

x=-4, y=3