How do I solve system of equation word problems.?

Question

Hi I’m having some trouble understanding system of equation word problems. I never set up the equation right and never get the right answer. Can someone give me an easier way on how to break down the problem and setting the equation up so I can understand how to do it and get the right answer

Answer 1

Try using the "V-E-S-T" method. Each letter represents one of the 4 steps you should perform.





Step 1 = Variables = Define your Variables (V)



We have two routines that take a certain amount of time. The question is asking us to find out how long each routine takes, so you can be pretty sure you'll need variables representing the time of each routine.



Let a be the time needed for the arm routine (in minutes)

Let b be the time needed for the abdominal routine (in minutes)





Step 2 - Equations = Set up Equations (E) for what you are told.



We are told that Keenan did 1 arm routine (1a) and 2 abdominal routines (2b) and that took 43 minutes.

a + 2b = 43



We are also told that he did 1 arm routine (1a) and 4 abdominal routines (4b) and that took 67 minutes

a + 4b = 67



Those are your two equations and since we have two unknowns, it should be solvable.





Step 3 - Solve = Solve (S) the system of equations



a + 2b = 43

a + 4b = 67



Given that you have the same coefficient (implied 1) in front of a in both equations, you can cancel out a but using "elimination". Subtract the 1st equation from the 2nd equation and the a will disappear.

a + 4b = 67

a + 2b = 43

----------------

..... 2b = 24



Divide both sides by 2:

b = 24/2

b = 12



So the abdominal routine takes 12 minutes.



Now use either equation to solve for a.

a + 2b = 43

a + 2(12) = 43

a + 24 = 43

a = 43 - 24

a = 19



Thus the arm routine takes 19 minutes.





Step 4 - Test = Test (T) your answers make sense.



1 arm routine + 2 abdominal routines should take 43 minutes

1(19) + 2(12) =? 43

19 + 24 =? 43

43 = 43 ✓



1 arm routine + 4 abdominal routines should take 67 minutes

1(19) + 4(12) =? 67

19 + 48 =? 67

67 = 67 ✓





Write down your answer in numbers and words. Be sure to look back to how you defined your variables.

a = 19, b = 12



Answer:

Arm routines take 19 minutes

Abdominal routines take 12 minutes

Answer 2

 Keenan is trying to incorporate more exercise into his busy schedule. 

 He has several short exercise routines he can complete at home. 

 

 Last week, he worked out for a total of 43 minutes 

 by doing 1 arm routine and 2 abdominal routines. 

 

 This week, he has completed 1 am routine and 4 abdominal routines 

 and spent a total of 67 minutes exercising. 

 How long does each routine last? 



 An arm routine takes 19 minutes to complete, 

 and an abdominal routine takes 12 minutes to complete.

Answer 3

I like to start out by defining my variables and give them a meaning:



Let r = time it takes to do one arm routine

Let d = time it takes to do one abdomen routine



If 1 arm and 2 ab routines take 43 min, we can create this equation:



r + 2d = 43



Another day he did 1 arm and 4 ab routines and it took 67 min, so:



r + 4d = 67



We now have a system of two equations and two unknowns that can be solved.



I'll use substitution.  Solving the first equation for r in terms of d:



r + 2d = 43

r = 43 - 2d



Substitute into the other equation and solve for d:



r + 4d = 67

43 - 2d + 4d = 67

2d = 24

d = 12



Now we can solve for r:



r = 43 - 2d

r = 43 - 2(12)

r = 43 - 24

r = 19



It takes him 19 min to do his arms routine and 12 minutes to do his abdominal routine.



I define my variables, including labels (minutes, in this case), break down each sentence, one at a time, to create equations that make sense (adding two minutes together to get minutes, etc.) and then solve the system of equations.



Hope this helped.