System of Linear Equations: a & b Values

In summary, the system given has one solution when a ≠ 0 and b ≠ -1/3, no solutions when a = 0 and b ≠ -1/3, and infinitely many solutions when a = 0 and b = -1/3. The row reduction process does not provide enough information to determine the values of a and b.
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Homework Statement



For the following system, indicate for what values of a and b the system will have
i.) One Solution
ii.) No Solutions
iii.) Infinitely Many Solutions

Homework Equations


2x + y - az = 1
5x + 3y - 2az = 2+2b
x + y + az = b

The Attempt at a Solution



[ 1 0 -2a | 1-b ]
[ 0 1 2a | 3b-1 ]
[ 0 0 a | -b ]

Every time I row reduce the system I end up with a row echelon form that leaves me with no means to decipher a and b. Making me think I'm doing something wrong.

Thanks.
 
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  • #2


I think you'll need to show use what you got for the row echelon reduction before anyone can comment.
 
  • #3


Yeah I just put it in the original post.

Thank you for the heads up.
 

FAQ: System of Linear Equations: a & b Values

What are the variables in a system of linear equations?

The variables in a system of linear equations are usually represented by x and y, although they can be represented by any letter. These variables represent the unknown quantities that we are trying to find in the equations.

How many equations are needed to solve a system of linear equations?

At least two equations are needed to solve a system of linear equations. This is because we need two independent equations to find the values of two variables. If there are more than two variables, we will need an equal number of equations to solve the system.

What are the possible solutions to a system of linear equations?

A system of linear equations can have one unique solution, no solution, or infinitely many solutions. The type of solution depends on the relationship between the equations. If the equations are parallel, there is no solution. If the equations are identical, there are infinitely many solutions. In all other cases, there is one unique solution.

How do you solve a system of linear equations with two variables?

To solve a system of linear equations with two variables, we can use the substitution method or the elimination method. In the substitution method, we solve one equation for one variable and substitute that value into the other equation. In the elimination method, we manipulate the equations to eliminate one variable and solve for the other. Both methods will result in the same solution.

What are the "a" and "b" values in a system of linear equations?

The "a" and "b" values in a system of linear equations represent the coefficients of the variables. In the equation ax + by = c, a and b are the coefficients of x and y, respectively. These values determine the slope and y-intercept of the line represented by the equation.

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