Solve using substitution
3x - 9y = 3
6x - 3y = -24
Solve using elimination
y - 1/2x = 6
2x + 6y = 19
divide first equation by 3
x - 3 y = 1
so
x = (3y+1)
substitute that for x in the second equation after dividing the second equation by 3
2 (3y+1) - y = -8
6 y + 2 - y = -8
5 y = -10
y = -2
then x -3(-2) = 1
x + 6 = 1
x = -5
y - 1/2x = 6
is
-x + 2 y = 12
2x + 6y = 19
multiply first eqn by 2 again
-2x + 4 y = 24
+2x + 6 y = 19
============== add
0 + 10 y = 43
y = 4.3
4.3 -.5 x = 6
.5 x = -1.7
x = - 3.4
To solve the system of equations using substitution, follow these steps:
1. Solve one of the equations for one variable in terms of the other.
Let's solve the first equation for x:
3x - 9y = 3
Add 9y to both sides:
3x = 9y + 3
Divide both sides by 3:
x = 3y + 1
2. Substitute the expression for x into the other equation.
Now, substitute the expression (3y + 1) for x in the second equation:
6x - 3y = -24
Replace x with (3y + 1):
6(3y + 1) - 3y = -24
Simplify the equation:
18y + 6 - 3y = -24
Combine like terms:
15y + 6 = -24
Subtract 6 from both sides:
15y = -30
Divide both sides by 15:
y = -2
3. Substitute the value of y back into either equation to solve for x.
Let's substitute y = -2 into the first equation:
3x - 9(-2) = 3
Simplify:
3x + 18 = 3
Subtract 18 from both sides:
3x = -15
Divide both sides by 3:
x = -5
Therefore, the solution to the system of equations is x = -5 and y = -2.
To solve the second system of equations using the elimination method, follow these steps:
1. Multiply one or both equations by appropriate constants to make the coefficients of one of the variables in both equations the same or additive inverses of each other.
Let's multiply the first equation by 2:
2(y - 1/2x) = 2(6)
Simplify:
2y - x = 12
2. Add or subtract the equations to eliminate one variable.
Now, subtract the modified first equation from the second equation:
2x + 6y - (2y - x) = 19 -12
Simplify:
2x - x + 6y - 2y = 7
Combine like terms:
x + 4y = 7
3. Solve the resulting equation for one variable.
Now, solve the equation x + 4y = 7 for x or y. Let's solve it for x:
x = 7 - 4y
4. Substitute the value of the solved variable back into one of the original equations and solve for the other variable.
Substitute the expression (7 - 4y) for x in the first equation:
y - 1/2(7 - 4y) = 6
Simplify:
y - 7/2 + 2y = 6
Combine like terms:
3y - 7/2 = 6
Add 7/2 to both sides:
3y = 6 + 7/2
Simplify:
3y = 19/2
Divide both sides by 3:
y = 19/2 * 1/3
y = 19/6
5. Substitute the value of the solved variable back into the other equation to solve for the remaining variable.
Substitute y = 19/6 into x = 7 - 4y:
x = 7 - 4(19/6)
Simplify:
x = 7 - 19/3
Multiply 7 by 3/3 to have the same denominator:
x = 21/3 - 19/3
Simplify:
x = 2/3
Therefore, the solution to the second system of equations is x = 2/3 and y = 19/6.