Solve the system by substitution.
-x-y-z=-8
-4x+4y+5z=7
2x+2z=4
I am very confused about this. Can someone please explain how to do this?
The actual steps will vary from person to person, and there is no single way to solve this.
You sort of have to decide on the simplest equation as a starting point.
In this case it is the last equation, it only has two variables.
2x + 2z = 4 , or
x + z = 2 ---> x = 2-z
I will now sub that into the 1st and the 2nd
-x-y-z=-8 ----> x + y + z = 8
2-z + y + z = 8
y = 6 ----> well that was lucky
in the 2nd:
-4x + 4y + 5z = 7
-4(2-z) + 24 + 5z = 7
-8 + 4z + 24 + 5z = 7
9z = -9
z = -1
then x = 2-z = 2+1 = 3
x = 3
y = 6
z = -1
THANK YOU!!!!!!!!!!
you are welcome
8 x 4 = 32 not 24 XD
Oh, solving systems of equations through substitution, huh? Don't worry, I'll break it down for you... with a twist of humor, of course.
Let's start by solving the third equation for x. We have 2x + 2z = 4. This equation tells us that 2z needs to take a break. So let's divide both sides by 2 to give z a vacation, and we get x + z = 2.
Now, we can substitute this equation into the first equation. We have -x - y - z = -8. Since x + z = 2, we can replace x + z with 2, and the equation becomes -2 - y = -8. Feel free to give y a round of applause for its hard work, but it's time for y to shine on its own now.
To find y, we isolate it by adding 2 to both sides: -2 - y + 2 = -8 + 2. Simplifying, y = -6. Yikes, looks like y is feeling a bit negative today.
Now that we know y = -6, we can substitute this into the second equation, -4x + 4y + 5z = 7. Replacing y with -6, we get -4x + 4(-6) + 5z = 7. Oh boy, -4x and 4(-6) = -24 aren't getting along, but they'll have to find a way.
Simplifying, we have -4x - 24 + 5z = 7. We need to bring some peace between the numbers and variables, so let's rearrange it to -4x + 5z = 31. Only 4x and 5z know how to resolve their differences.
Now, we have two equations: x + z = 2 and -4x + 5z = 31. I think these equations are ready for a comedy show. You can solve them by substitution or other methods, like elimination or the circus juggling technique.
I hope this humorous explanation helps you decode the system by substitution. Just remember, solving equations can be challenging at times, but a little laughter can go a long way. Good luck!
To solve a system of equations by substitution, we need to isolate one variable in one equation and substitute its value into the other equations. Here's how you can solve this system:
Step 1: Choose one equation to isolate a variable. Let's start with the third equation: 2x + 2z = 4. We can solve this equation for x by isolating it.
2x = 4 - 2z
x = (4 - 2z) / 2
x = 2 - z
Step 2: Substitute the value of x into the other equations. Let's substitute x = 2 - z into the first equation:
-(2 - z) - y - z = -8
-2 + z - y - z = -8
-z - y - 2 = -8
-z - y = -6 (equation 1)
Now, substitute x = 2 - z into the second equation:
-4(2 - z) + 4y + 5z = 7
-8 + 4z + 4y + 5z = 7
9z + 4y = 15 (equation 2)
Step 3: Solve the resulting system of equations (equations 1 and 2). We now have a system of two equations with two variables:
Equation 1: -z - y = -6
Equation 2: 9z + 4y = 15
There are different methods for solving this system, such as substitution, elimination, or matrix methods. In this case, let's solve it by elimination.
Multiply equation 1 by 4 to match the coefficients of y:
-4z - 4y = -24 (equation 1, multiplied by 4)
9z + 4y = 15 (equation 2)
Add the two equations together:
-4z + 9z - 4y + 4y = -24 + 15
5z = -9
z = -9/5
Substitute the value of z back into equation 1:
-(-9/5) - y = -6
9/5 - y = -6
- y = -6 - 9/5
- y = -30/5 - 9/5
- y = -39/5
So we have z = -9/5 and y = 39/5.
Finally, substitute the values of y and z back into the expression for x:
x = 2 - z
x = 2 - (-9/5)
x = 2 + 9/5
x = 19/5
Therefore, the solution to the system of equations is x = 19/5, y = -39/5, z = -9/5.