The question in the book reads:
"Suppose you try to solve a system of linear equations using substitution and get this result. How many solutions does each system have?"
1)x=-2
2)y=12
3)6=x
4)-8=-8
5)7=y
6)5=3
I understand #3 and #6, but I do not understand how to know how many solutions there are to #1,2,3,or 5.
1)one solution
2)one solution
4)infinite solutions
5)one solution
To determine the number of solutions for each system of linear equations, we need to understand the concept of a solution in the context of substitution.
When solving a system of linear equations using substitution, we substitute the value of one variable from one equation into the other equation, which helps us find the value of the other variable. We repeat this process until we can find the values of both variables. The goal is to find a unique solution (one solution), an infinite number of solutions, or no solution.
Let's analyze each system of equations:
1) x = -2
This equation tells us that the value of x is always -2, regardless of the value of y. In this case, we have an infinite number of solutions. This is because for any value of y, the equation x = -2 is always satisfied.
2) y = 12
Similarly, this equation tells us that the value of y is always 12, regardless of the value of x. Again, we have an infinite number of solutions, as any value of x will satisfy this equation.
3) 6 = x
Here, we have a single equation with one variable. From this equation, we see that x is always equal to 6. Hence, the system has a unique solution, as there is only one possible pair of x and y that will satisfy this equation.
4) -8 = -8
This equation is a tautology, meaning it is always true regardless of the values of x and y. Therefore, any values of x and y will satisfy this equation. As a result, we have an infinite number of solutions for this system.
5) 7 = y
Similar to the previous equation, we have a single equation with one variable. From this equation, we find that y is always equal to 7. Therefore, the system has a unique solution.
6) 5 = 3
This equation is a contradiction, as 5 cannot be equal to 3. Hence, there are no values of x and y that will satisfy this equation. Therefore, this system has no solution.
In summary:
- Systems 1 and 2 have an infinite number of solutions.
- Systems 3 and 5 have a unique solution.
- Systems 4 and 6 have no solution.