Solve using substitution
3x - 9y = 3
6x - 3y = -24
Solve using elimination
y - 1/2x = 6
2x + 6y = 19
3x - 9y = 3
6x - 3y = -24
from the first, divide by 3
x - 3y = 1
x = 3y+1
sub into the 2nd
6(3y+1) - 3y = -24
18y + 6 - 3y = -24
15y = -30
y = -2
then x = -6+1 = -5
for the 2nd system, multiply the 1st by -4
2x -4y = -24
2x + 6y = 19
subtract them
10y = 43
y = 43/10 = 4.3
into the 2nd:
2x + 6(4.3) = 19
2x = 19 - 25.8
2x = -6.8
x = -3.4
Eq1: 3x - 9y = 3
Eq2: 6x - 3y = -24
In Eq1, solve for X:
3x = 9y + 3
X = 3y+1
In Eq2, replace X with 3y+1:
6(3y+1) - 3y = -24
18y+6 - 3y = -24
15y = -30
Y = -2.
In Eq1, replace Y with-2:
3x-9*(-2) = 3
3x + 18 = 3
3x = -15
X = -5.
Solution set: (-5,-2).
Eq1: Y - X/2 = 6
Eq2: 2x + 6y = 19
Multiply Eq1 by 4 and add Eq2:
-2x + 4y = 24
+2x + 6y = 19
Sum: 10y = 43
Y = 4.3
In Eq1, replace Y with 4.3:
4.3 - 0.5x = 6
-0.5x = 1.7
X = -3.4
Solution set: (-3.4,4.3).
Reiny, we meet again!! LOL! Oh well, we
got the same answer!
To solve the first set of equations using substitution, you can follow these steps:
Step 1: Solve one equation for one variable in terms of the other variable.
In this case, let's solve the first equation for x:
3x - 9y = 3
3x = 9y + 3
x = (9y + 3) / 3
x = 3y + 1
Step 2: Substitute the expression for the solved variable into the other equation.
Substitute x = 3y + 1 into the second equation:
6x - 3y = -24
6(3y + 1) - 3y = -24
18y + 6 - 3y = -24
15y + 6 = -24
Step 3: Solve the resulting equation for the remaining variable.
15y + 6 = -24
15y = -24 - 6
15y = -30
y = -30 / 15
y = -2
Step 4: Substitute the value of y back into either equation to find the value of the other variable.
Using the first equation:
3x - 9y = 3
3x - 9(-2) = 3
3x + 18 = 3
3x = 3 - 18
3x = -15
x = -15 / 3
x = -5
Therefore, the solution to the system of equations is x = -5 and y = -2.
To solve the second set of equations using elimination, you can follow these steps:
Step 1: Multiply one or both equations by a factor if necessary to create opposite coefficients for one of the variables.
Let's multiply the first equation by 2 to eliminate the fractions:
2(y - 1/2x) = 2(6)
y - x = 12
Step 2: Add or subtract the equations to eliminate a variable.
(y - x) + (2x + 6y) = 12 + 19
y + 2x - x + 6y = 31
7y + x = 31
Step 3: Solve the resulting equation for one variable.
7y + x = 31
x = 31 - 7y
Step 4: Substitute the value of the solved variable into either original equation to find the other variable.
Using the first equation:
y - 1/2x = 6
y - 1/2(31 - 7y) = 6
y - 15.5 + 3.5y = 6
4.5y = 21.5
y = 21.5 / 4.5
y ≈ 4.78
Step 5: Substitute the value of y back into the expression for x to find its value.
x = 31 - 7y
x = 31 - 7(4.78)
x = 31 - 33.46
x ≈ -2.46
Therefore, the solution to the system of equations is x ≈ -2.46 and y ≈ 4.78.