system of equations thank you
2x+3y-17
3x+y=31
2(x-3y)=24
y=3x-2
2(x+4)=24
x+y=17
I will do the second:
2x-6y=24
-3x+y=-2
multipy the second equation by 6 to eliminate the y's.
2x-6y=24
-18x+6y=-12
add the equations
-16x=12
x=-3/4
then solve for y by putting that x back into one of the equations.
I will be happy to check your work on the others.
i don't think that's write cuz they show me another way
To solve a system of equations, you can use various methods such as substitution, elimination, or graphing. I'll explain how to solve each of the given systems using substitution method.
System 1:
Equation 1: 2x + 3y = 17
Equation 2: 3x + y = 31
Step 1: Solve Equation 2 for one variable in terms of the other.
From Equation 2, we can isolate y by subtracting 3x from both sides:
y = 31 - 3x
Step 2: Substitute the expression of y from Equation 2 into Equation 1.
Replace y in Equation 1 with 31 - 3x:
2x + 3(31 - 3x) = 17
Step 3: Simplify and solve for x.
Distribute 3 to each term within the parentheses:
2x + 93 - 9x = 17
Combine the x-terms:
-7x + 93 = 17
Subtract 93 from both sides:
-7x = 17 - 93
-7x = -76
Divide by -7:
x = -76 / -7
x ≈ 10.857
Step 4: Substitute the value of x back into Equation 2 to find y.
Using Equation 2: y = 31 - 3x
Substituting x = 10.857:
y = 31 - 3(10.857)
y = 31 - 32.571
y ≈ -1.571
Therefore, the solution to System 1 is x ≈ 10.857 and y ≈ -1.571.
System 2:
Equation 1: 2(x - 3y) = 24
Equation 2: y = 3x - 2
Step 1: Solve Equation 2 for one variable in terms of the other.
Equation 1 does not need solving because it is already in terms of x.
Step 2: Substitute the expression from Equation 2 into Equation 1.
Replace y in Equation 1 with 3x - 2:
2(x - 3(3x - 2)) = 24
Step 3: Simplify and solve for x.
Distribute 3 to each term within the inner parentheses:
2(x - 9x + 6) = 24
Combine like terms within the parentheses:
2(-8x + 6) = 24
Distribute 2 to the terms inside the outer parentheses:
-16x + 12 = 24
Subtract 12 from both sides:
-16x = 24 - 12
-16x = 12
Divide by -16:
x = 12 / -16
x = -0.75
Step 4: Substitute the value of x back into Equation 2 to find y.
Using Equation 2: y = 3x - 2
Substituting x = -0.75:
y = 3(-0.75) - 2
y = -2.25 - 2
y = -4.25
Therefore, the solution to System 2 is x = -0.75 and y = -4.25.
System 3:
Equation 1: 2(x + 4) = 24
Equation 2: x + y = 17
Step 1: Solve Equation 1 for x.
Distribute 2 to the terms inside the parentheses:
2x + 8 = 24
Subtract 8 from both sides:
2x = 24 - 8
2x = 16
Divide by 2:
x = 16 / 2
x = 8
Step 2: Substitute the value of x back into Equation 2 to find y.
Using Equation 2: x + y = 17
Substituting x = 8:
8 + y = 17
Subtract 8 from both sides:
y = 17 - 8
y = 9
Therefore, the solution to System 3 is x = 8 and y = 9.