how do you solve the following questions using the substitution method?
-3x-y=3 3x+6y=48
-3x-5y=-21 -5x+6y=32
2x+4y=12
4x+16y=8
Substitution? I would solve for x in the first equation:
x=6-2y check that. Then put that for x into the second equation...
4(6-2y)+16y=8 and solve for y.
To solve the given system of equations using the substitution method, follow these steps:
1. Solve one equation for one variable
Choose one of the equations and isolate one of the variables in terms of the other variable.
For the first system of equations:
-3x - y = 3 ... Equation 1
3x + 6y = 48 ... Equation 2
From Equation 1, isolate y:
y = -3x + 3
For the second system of equations:
-3x - 5y = -21 ... Equation 3
-5x + 6y = 32 ... Equation 4
From Equation 3, isolate y:
y = (-3x + 21)/5
For the third system of equations:
2x + 4y = 12 ... Equation 5
4x + 16y = 8 ... Equation 6
From Equation 5, isolate y:
y = (12 - 2x)/4
2. Substitute
Take the expression for the isolated variable from Step 1 and substitute it into the other equation.
For the first system:
Substitute y = -3x + 3 into Equation 2:
3x + 6(-3x + 3) = 48
Simplify and solve for x:
3x - 18x + 18 = 48
-15x = 48 - 18
-15x = 30
x = 30/(-15)
x = -2
Now substitute the value of x back into one of the original equations, Equation 1, to solve for y:
-3(-2) - y = 3
6 - y = 3
-y = 3 - 6
-y = -3
y = (-3)/(-1)
y = 3
So the solution to the first system of equations is x = -2 and y = 3.
For the second system:
Substitute y = (-3x + 21)/5 into Equation 4:
-5x + 6((-3x + 21)/5) = 32
Simplify and solve for x:
-5x - 18x + 126 = 160
-23x = 160 - 126
-23x = 34
x = 34/(-23)
x = -34/23
Now substitute the value of x back into one of the original equations, Equation 3, to solve for y:
-3((-34)/23) - 5y = -21
(-102)/23 - 5y = -21
-102 - 115y = -483
-115y = -483 + 102
-115y = -381
y = (-381)/(-115)
y = 381/115
So the solution to the second system of equations is x = -34/23 and y = 381/115.
For the third system:
Substitute y = (12 - 2x)/4 into Equation 6:
4x + 16((12 - 2x)/4) = 8
Simplify and solve for x:
4x + 48 - 8x = 8
-4x = 8 - 48
-4x = -40
x = (-40)/(-4)
x = 10
Now substitute the value of x back into one of the original equations, Equation 5, to solve for y:
2(10) + 4y = 12
20 + 4y = 12
4y = 12 - 20
4y = -8
y = (-8)/4
y = -2
So the solution to the third system of equations is x = 10 and y = -2.
Therefore, the solutions for the three given systems of equations using the substitution method are:
System 1: x = -2 and y = 3
System 2: x = -34/23 and y = 381/115
System 3: x = 10 and y = -2