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Describing Solutions to Systems of Linear Equations

Describing Solutions to Systems of Linear Equations 1.6 - Solution

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a

We want to find the number of solutions for the following system of equations. To determine how many solutions this system has, we will solve it by substitution. Doing so will result in one of three cases.

Result of solving by substitution Number of solutions
A value for and is determined. One solution
An identity is found, such as Infinitely many solutions
A contradiction is found, such as No solution

Thus, we need to solve the system of equations before we can make our conclusion. When solving a system of equations using substitution, there are three steps.

  1. Isolate a variable in one of the equations.
  2. Substitute the expression for that variable into the other equation and solve.
  3. Substitute this solution into one of the equations and solve for the value of the other variable.
Let's isolate in Equation (I). Let's substitute for in Equation (II) and solve the resulting equation for
Solving this system resulted in a contradiction. This means that the system has no solution.
b

Here we have been asked to determine how many solutions the system has. We will first solve the system by substitution. After that we will use the following table to find the number of solutions.

Result of solving by substitution Number of solutions
A value for and is determined. One solution
An identity is found, such as Infinitely many solutions
A contradiction is found, such as No solution
Let's begin the solution by isolating in Equation (II). Next step of the solution is to substitute for in Equation (I) and solve the resulting equation.
Let's continue by substituting for in Equation (II) and solve the resulting equation for
The solution of the system resulted in us finding a value for and a value for Thus, the given system of equations has one solution.


c
We have been given the following system of equations.

Now we want to find out how many solutions the system has. We will do that by first solving the system using the substitution method. After that we will with help of this table determine how many solutions the system has.

Result of solving by substitution Number of solutions
A value for and is determined. One solution
An identity is found, such as Infinitely many solutions
A contradiction is found, such as No solution
First step of the solution is to isolate one variable in one of the equations. Let's isolate in Equation (II). We can now find by substituting for in Equation (I).
Here we ended up with an identity. Therefore, we know that the system has an infinite number of solutions.