ACT , Algebra 1 – system of equations

MathPro Q&A Forum (ACT)

 Q: For what value of A will the following system of equations have infinitely many solutions?

3x – 2y = 4

Ax + 4y = -8

 

Answer: -6

Explanation: In order for the system to have infinitely many solutions, the two equations must describe the same line.  (Recall that the solution(s) of a system are the points where the graphs of the two lines meet.  For the two lines to meet in infinitely many points, they must be the same line.)

 To make the second equation the same as the first, multiply both sides of the first equation by -2:

 -2(3x – 2y) = -2(4)

-6x + 4y = -8

So, A = -6.

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Algebra 1, 2 – systems of equations

Q: Red pens cost twice as much as blue pens. Two blue pens and three red pens cost a total of $2.32. How much does a blue pen cost?

 Answer:  $0.29

 Explanation:  Let r be the cost of a red pen, and let b be the cost of a blue pen. 

Then, r = 2b and 2b + 3r = 2.32. Solve this system of equations. 

By substitution,

r + 3r = 2.32

4r = 2.32

r = 0.58

Plug r= 0.58 into the first equation (or second, but I like the first better):

0.58 = 2b

0.29 = b

 You can also use other algebraic methods such as elimination, Cramer’s Rule, etc.  Or use my TI-84 Plus program CRAMER once you rearrange the system to say: 2b – r = 0 and 2b + 3r = 2.32.

 

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SAT – systems of equations

Q: A phone call costs c cents for the first 10 minutes and n cents per minute for additional minutes. A 20-minute phone call costs 50 cents, and a 40-minute phone call costs 90 cents. What is the value of c?
 
Answer: 30
Explanation:  A 20-minute call costs c cents for the first 10 minutes and n cents per minute for each of the next 10 minutes.  So, c+10n=50.  A 40-minute call costs c cents for the first 10 minutes and n cents per minute for each of the next 30 minutes: c+30n=90.  Solve this system of equations.  Use my TI-84 Plus program CRAMER or an algebraic method such as substitution, elimination, Cramer’s Rule, etc.  c=30.
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Algebra 1, 2 – systems of equations

Q: 100 people are to be seated at 22 tables. Some tables seat 4 people and some seat 6 people. If no seats are empty, how many tables seating 4 people are there?
 
Answer: 16
Explanation:  Solve the system: x + y = 22; 4x + 6y = 100.  My ‘CRAMER’ program will solve this for you! The number of 4-person tables is the x-value of the solution.
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Algebra 1, 2 – systems of equations

Two cubes and one pyramid weigh 9 lb. Seven cubes and two pyramids weigh 24 lb. All cubes weigh the same, and all pyramids weigh the same. What is the weight of one pyramid?

 
Answer: 5 lb

Explanation:  Let x be the weight of a cube.  Let y be the weight of a pyramid. 

The first sentence becomes: 2x + y = 9. 

The second sentence is: 7x + 2y = 24. 

Solve this as a system of equations by any method you choose… substitution, elimination, Cramer’s Rule, etc.  Or use my TI-84 Plus program ‘CRAMER’.

x = 2, y = 5

‘y’ is the weight of one pyramid, so the answer is 5.

 

 

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