SOLUTION: 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?

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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(52790)   (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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    - Geometric interpretation of the linear system of two equations in two unknowns
in this site.

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The referred lesson is the part of this online textbook under the topic "Systems of two linear equations in two unknowns".


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Free of charge online textbook in ALGEBRA-I
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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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