Solve the system by substitution
2x-y+z=-4
z=5
-2x+3y-z=-10
I think the answer is (-8,-7,5)
But can someone tel me the right one and explain I’m so lost
kk is correct
Check: when (-8,-7, 5)
2x-y+z=-4
2(-8) -(-7) + 5 = -4
-16 +7+5=-4
-16+12=-4
-4=-4
Oh, don't be lost, I am here to assist you! Let's solve this system by substitution step-by-step, and I'll try to explain it in a way that takes away the confusion.
First, we have the equation z = 5. This equation is already solved for z, so we can substitute 5 for z in the other two equations. Let's do that now:
Equation 1: 2x - y + z = -4
Substituting z = 5: 2x - y + 5 = -4
Equation 2: -2x + 3y - z = -10
Substituting z = 5: -2x + 3y - 5 = -10
Now, we have a system of two equations with two variables (x and y):
Equation 1: 2x - y + 5 = -4
Equation 2: -2x + 3y - 5 = -10
To solve this system, we can solve one equation for one variable and substitute the result into the other equation. Let's solve Equation 1 for y:
2x - y + 5 = -4
-y = -2x - 9 (subtracting 5 from both sides)
y = 2x + 9 (multiply both sides by -1)
Now, we substitute y = 2x + 9 into Equation 2:
-2x + 3(2x + 9) - 5 = -10
-2x + 6x + 27 - 5 = -10
4x + 22 = -10
4x = -32
x = -8
We've found the value of x, which is -8. Now, we can substitute this value back into y = 2x + 9 to find y:
y = 2(-8) + 9
y = -16 + 9
y = -7
So, the solution for this system of equations is x = -8, y = -7. Now, let's substitute these values into the given equation z = 5 to check if it holds:
z = 5 (given)
z = 5
Great! The value of z remains the same. Therefore, the solution for this system of equations is indeed (-8, -7, 5). Keep in mind that the process may seem confusing at first, but with practice, you'll get the hang of it!
To solve the system of equations by substitution, we can follow these steps:
Step 1: Solve one equation for one variable.
Let's solve the third equation for z:
-2x + 3y - z = -10
We can rearrange the equation to isolate z:
z = -2x + 3y + 10
Step 2: Substitute the expression for the variable into the other equations.
Substitute z = 5 into the first equation:
2x - y + z = -4
2x - y + 5 = -4
Simplify the equation:
2x - y = -9
Step 3: Solve the resulting equation.
We now have two equations:
2x - y = -9 (Equation 1)
2x - y + 5 = -4 (Equation 2)
We can use Equation 1 to solve for one variable. Let's solve for y:
-y = -9 - 2x
y = 9 + 2x (Equation 3)
Step 4: Substitute Equation 3 into Equation 2.
We substitute Equation 3 into Equation 2 and solve for x:
2x - (9 + 2x) + 5 = -4
Simplify the equation:
2x - 9 - 2x + 5 = -4
Combine like terms:
-4 = -4
The resulting equation shows that -4 is equal to -4, which is a true statement. This means that x can take on any value.
Step 5: Solve for y using x.
Substitute x into Equation 3 to solve for y:
y = 9 + 2x
Since x is a free variable, we can assign any value to it to find the corresponding value of y.
Step 6: Solve for z using x and y.
Substitute x and y into the second equation:
z = -2x + 3y + 10
Again, since x and y are free variables, we can assign any values to them to find the corresponding value of z.
In conclusion, the solution to the system of equations will have infinitely many solutions with the form (x, y, z) = (x, 9 + 2x, -2x + 3(9 + 2x) + 10). Therefore, there is not a single correct answer like (-8, -7, 5).
2x - y + 5 = -4 ... 2x - y = -9
-2x + 3y - 5 = -10 ... -2x + 3y = -5
adding equations (to eliminate x)
... 2y = -14