Use the substitution method to solve the linear system.
1/8p+3/4q=7
3/2p-q=4
Thanks.
1st times 8 ----> p + 6q = 56 or p = 56-6q
2nd times 2 ---> 3p - 2q = 8
3(56-6q) - 2q = 8
168 - 18q - 2q = 8
-20q = -160
q = 8
then p = 56-6(8) = 8
p= 8 and q = 8
Notice how for both of your last two posts, I got rid of fractions to make the equations much more manageable.
To solve the linear system using the substitution method, follow these steps:
Step 1: Solve one equation for one variable
Choose either equation to solve for one variable in terms of the other variable. Let's choose the second equation:
3/2p - q = 4
Rearrange the equation to isolate one variable. Add q to both sides:
3/2p = q + 4
Step 2: Substitute the expression from step 1 into the other equation
Now, substitute the expression for q from step 1 into the first equation:
1/8p + 3/4(q) = 7
Replace q with the expression (q + 4) in the first equation:
1/8p + 3/4(q + 4) = 7
Step 3: Solve the resulting equation
Multiply both sides of the equation by 8 to clear the fraction:
p + 6(q + 4) = 56
Distribute the multiplication:
p + 6q + 24 = 56
Combine like terms:
p + 6q = 56 - 24
p + 6q = 32
Step 4: Solve the resulting linear equation
Now, we have a new linear equation:
p + 6q = 32
We can use this equation to solve for one variable in terms of the other. Let's solve for p:
p = 32 - 6q
Step 5: Substitute the value of p or q into one of the original equations
Now that we have an expression for p, we can substitute it into one of the original equations. Let's use the second equation:
3/2p - q = 4
Replace p with the expression (32 - 6q) in the second equation:
3/2(32 - 6q) - q = 4
Step 6: Solve the resulting equation
Distribute the multiplication:
48 - 9q - q = 4
Combine like terms:
48 - 10q = 4
Subtract 48 from both sides:
-10q = 4 - 48
-10q = -44
Divide both sides by -10 to solve for q:
q = -44 / -10
q = 4.4
Step 7: Find the value of the other variable
Substitute the value of q back into one of the original equations. Let's use the first equation:
1/8p + 3/4q = 7
Replace q with 4.4 in the first equation:
1/8p + 3/4(4.4) = 7
Simplify:
1/8p + 33/10 = 7
Multiply both sides of the equation by 8 to clear the fraction:
p + 26.4 = 56
Subtract 26.4 from both sides:
p = 56 - 26.4
p = 29.6
Step 8: Final Solution
The solution to the linear system is p = 29.6 and q = 4.4.
Thus, the solution to the given linear system using the substitution method is p = 29.6 and q = 4.4.