Question 1097061: 2/3y + 1/4x = 2
8y + 3x = a
in this system of equations a is a constant.If the system has infinite solutions, what is the value of a?
Found 3 solutions by ikleyn, josh_jordan, greenestamps:
Answer by ikleyn(52788)   (Show Source):
You can put this solution on YOUR website! .
The coefficients of the first equation are PROPORTIONAL to the coefficients of the second equation with the proportionality coefficient of  .
Then the necessary and sufficient condition for the system to have a solution is that the right side terms are proportional with the same coefficient.
It gives an equation for the term "a":  = 2, which gives a = 24.
At the same time, this condition is a necessary and sufficient for the system to have INFINITE number of solutions.
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See the lesson
- Geometric interpretation of the linear system of two equations in two unknowns
in this site.
Also, you have this free of charge online textbook in ALGEBRA-I in this site
- ALGEBRA-I - YOUR ONLINE TEXTBOOK.
The referred lesson is the part of this online textbook under the topic "Systems of two linear equations in two unknowns".
Save the link to this online textbook together with its description
Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson
to your archive and use it when it is needed.
Answer by josh_jordan(263)  (Show Source):
You can put this solution on YOUR website! In order for a system to have infinitely many solutions, you must have a situation where, when you add the two equations in the system together, your result is 0 = 0. (zero on the left side of the equation and zero on the right)
To find the value of a so that the system has infinite solutions, let's start by getting rid of the fractions from equation 1 in our system. To do this, we can multiply the entire equation by 12, which is the least common denominator. This gives us
8y + 3x = 24
Notice how the left side of equation 1 now matches the left side of equation 2?
8y + 3x = 24
8y + 3x = a
This tells us that in order for both equations to match, "a" MUST be 24. So, if a is 24, let's try to solve our system. What will happen?
First, we can multiply equation 1 by -1, giving us
-8y - 3x = -24
8y + 3x = 24
Adding equation 1 and 2 together, we get:
0 = 0
Since 0 = 0, this system of linear equations has infinitely many solutions when a = 24.
Answer: a = 24
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
For a system of two linear equation to have an infinite number of solutions, the two equations must represent the same line. Algebraically, that means each equation is a multiple of the other.
In the given pair of equations, the coefficient of y in the second equation is 12 times the coefficient of y in the first equation (2/3 times 12 equals 8), and the coefficient of x in the second equation is 12 times the coefficient of x in the first equation (1/4 times 12 equals 3). So the constant in the second equation has to be 12 times the constant in the first equation: 2*12 = 24.
Answer: a = 24
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