The solution set of the linear system…
is the set of all possible solutions.
In order to find the solution set of a linear system, we need to solve for the variables using the given equations.
A linear system is a set of two or more linear equations that contain two or more variables. For example, consider the following linear system:
2x + 3y = 7 (equation 1)
5x – y = 8 (equation 2)
To find the solution set of this linear system, we will need to solve for x and y. We can do this by using either substitution or elimination method.
Substitution Method:
1. From equation 2, solve for y:
y = 5x – 8
2. Substitute this expression for y into equation 1 and solve for x:
2x + 3(5x – 8) = 7
2x + 15x – 24 = 7
17x = 31
x = 31/17
3. Substitute this value of x into either equation 1 or 2 and solve for y:
2(31/17) + 3y = 7
3y = (119/17) – (62/17)
y = 3/17
Therefore, the solution set for the linear system is (31/17, 3/17).
Elimination Method:
1. Multiply equation 2 by 3:
15x – 3y = 24
2. Add equation 1 and the new equation from step 1 to eliminate y:
2x + 3y + 15x – 3y = 7 + 24
17x = 31
x = 31/17
3. Substitute this value of x into either equation 1 or 2 and solve for y:
2(31/17) + 3y = 7
3y = (119/17) – (62/17)
y = 3/17
Therefore, the solution set for the linear system is (31/17, 3/17).
In summary, to find the solution set of a linear system, we need to solve for all the variables in the system using either substitution or elimination method. The solution set will be the set of values for the variables that make all the equations in the system true.
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