How to I solve this linear system using method of substitution?
1/4x + 1/2y = -5/12
2x - y = -4/3
To solve a linear system using the method of substitution, follow these steps:
Step 1: Solve one of the equations for one variable in terms of the other variable.
Let's solve the second equation for x.
2x - y = -4/3
Add y to both sides:
2x = y - 4/3
Divide by 2:
x = (y - 4/3)/2
Step 2: Substitute the expression solved for in Step 1 into the other equation.
Now, substitute (y - 4/3)/2 for x in the first equation:
1/4((y - 4/3)/2) + 1/2y = -5/12
Step 3: Solve the resulting equation for the remaining variable.
Now, you can solve the equation for y:
(1/4)(y - 4/3)/2 + 1/2y = -5/12
To simplify this equation, multiply both sides by 4:
(y - 4/3) + 2y = -20/12
y - 4/3 + 2y = -20/12
3y - 4 + 6y = -20/12
Combine like terms:
9y - 4 = -20/12
Now, add 4 to both sides:
9y = -20/12 + 4
9y = -20/12 + 48/12
9y = 28/12
Finally, divide by 9:
y = (28/12) / 9
y = 7/27
Step 4: Substitute the value of y back into one of the original equations and solve for the other variable.
We'll use the second equation to find x:
2x - y = -4/3
Substitute y = 7/27:
2x - (7/27) = -4/3
Now, add (7/27) to both sides:
2x = -4/3 + 7/27
To make the fractions have the same denominator, multiply -4/3 by 9/9 and multiply 7/27 by 3/3:
2x = (-36/27) + (21/27)
2x = -15/27
Simplify the equation further:
2x = -5/9
Finally, divide by 2 to solve for x:
x = (-5/9) / 2
x = -5/18
Therefore, the solution to the linear system is:
x = -5/18
y = 7/27
To do substitution, choose one of the equations. Then choose a variable in that equation, and express it in terms of the other variable. Finally, substitute it to the other equation.
Here, let's choose the 2nd equation.
2x - y = -4/3
Then let's choose the variable y. Express it in terms of the other:
2x - y = -4/3
-y = -4/3 - 2x
y = 2x + 4/3
Now, substitute this to the 1st equation:
1/4x + 1/2y = -5/12
1/4x + 1/2(2x + 4/3) = -5/12
1/4x + x + 2/3 = -5/12
5/4x = -5/12 - 2/3
5/4x = -5/12 - 8/12
5/4x = -13/12
x = -13/12 * 4/5
x = -13/15
Now that you have a value for x, substitute this to either equation to get the value of y.
Hope this helps~ `u`